Calculus Examples

$\int \frac{x^{2}}{\sqrt[3]{1 + 2 x}} d x$

Step 1

Let $u_{1} = 1 + 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 1 + 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $1 + 2 x$ .

$\frac{d}{d x} [1 + 2 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + 2 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [2 x]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [2 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$0 + 2 \frac{d}{d x} [x]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 + 2 \cdot 1$

Step 1.1.3.3

Multiply $2$ by $1$ .

$0 + 2$

Step 1.1.4

Add $0$ and $2$ .

$2$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{{(\frac{u_{1}}{2} - \frac{1}{2})}^{2}}{\sqrt[3]{u_{1}}} \cdot \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{{(\frac{u_{1}}{2} - \frac{1}{2})}^{2}}{\sqrt[3]{u_{1}}}$ by $\frac{1}{2}$ .

$\int \frac{{(\frac{u_{1}}{2} - \frac{1}{2})}^{2}}{\sqrt[3]{u_{1}} \cdot 2} d u_{1}$

Step 2.2

Move $2$ to the left of $\sqrt[3]{u_{1}}$ .

$\int \frac{{(\frac{u_{1}}{2} - \frac{1}{2})}^{2}}{2 \sqrt[3]{u_{1}}} d u_{1}$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{{(\frac{u_{1}}{2} - \frac{1}{2})}^{2}}{\sqrt[3]{u_{1}}} d u_{1}$

Step 4

Apply basic rules of exponents.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u_{1}}$ as ${u_{1}}^{\frac{1}{3}}$ .

$\frac{1}{2} \int \frac{{(\frac{u_{1}}{2} - \frac{1}{2})}^{2}}{{u_{1}}^{\frac{1}{3}}} d u_{1}$

Step 4.2

Move ${u_{1}}^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} \int {(\frac{u_{1}}{2} - \frac{1}{2})}^{2} {({u_{1}}^{\frac{1}{3}})}^{- 1} d u_{1}$

Step 4.3

Multiply the exponents in ${({u_{1}}^{\frac{1}{3}})}^{- 1}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \int {(\frac{u_{1}}{2} - \frac{1}{2})}^{2} {u_{1}}^{\frac{1}{3} \cdot - 1} d u_{1}$

Step 4.3.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{1}{2} \int {(\frac{u_{1}}{2} - \frac{1}{2})}^{2} {u_{1}}^{\frac{- 1}{3}} d u_{1}$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{2} \int {(\frac{u_{1}}{2} - \frac{1}{2})}^{2} {u_{1}}^{- \frac{1}{3}} d u_{1}$

Step 5

Let $u_{2} = \frac{u_{1}}{2} - \frac{1}{2}$ . Then $d u_{2} = \frac{1}{2} d u_{1}$ , so $2 d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 5.1

Let $u_{2} = \frac{u_{1}}{2} - \frac{1}{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 5.1.1

Differentiate $\frac{u_{1}}{2} - \frac{1}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2} - \frac{1}{2}]$

Step 5.1.2

By the Sum Rule, the derivative of $\frac{u_{1}}{2} - \frac{1}{2}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [\frac{u_{1}}{2}] + \frac{d}{d u_{1}} [- \frac{1}{2}]$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}] + \frac{d}{d u_{1}} [- \frac{1}{2}]$

Step 5.1.3

Evaluate $\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$ .

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Step 5.1.3.1

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{2}$ with respect to $u_{1}$ is $\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [- \frac{1}{2}]$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$\frac{1}{2} \cdot 1 + \frac{d}{d u_{1}} [- \frac{1}{2}]$

Step 5.1.3.3

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} + \frac{d}{d u_{1}} [- \frac{1}{2}]$

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

Since $- \frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $- \frac{1}{2}$ with respect to $u_{1}$ is $0$ .

$\frac{1}{2} + 0$

Step 5.1.4.2

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2}$

Step 5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} \frac{1}{\frac{1}{2}} d u_{2}$

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{1}{2} \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} (1 \cdot 2) d u_{2}$

Step 6.2

Multiply $2$ by $1$ .

$\frac{1}{2} \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} \cdot 2 d u_{2}$

Step 6.3

Move $2$ to the left of ${u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}}$ .

$\frac{1}{2} \int 2 {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2}$

Step 7

Since $2$ is constant with respect to $u_{2}$ , move $2$ out of the integral.

$\frac{1}{2} (2 \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2})$

Step 8

Simplify.

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Step 8.1

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2}$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Cancel the common factor.

$\frac{2}{2} \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2}$

Step 8.2.2

Rewrite the expression.

$1 \int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2}$

Step 8.3

Multiply $\int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2}$ by $1$ .

$\int {u_{2}}^{2} {(2 u_{2} + 1)}^{- \frac{1}{3}} d u_{2}$

Step 9

Let $u_{3} = 2 u_{2} + 1$ . Then $d u_{3} = 2 d u_{2}$ , so $\frac{1}{2} d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 9.1

Let $u_{3} = 2 u_{2} + 1$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 9.1.1

Differentiate $2 u_{2} + 1$ .

$\frac{d}{d u_{2}} [2 u_{2} + 1]$

Step 9.1.2

By the Sum Rule, the derivative of $2 u_{2} + 1$ with respect to $u_{2}$ is $\frac{d}{d u_{2}} [2 u_{2}] + \frac{d}{d u_{2}} [1]$ .

$\frac{d}{d u_{2}} [2 u_{2}] + \frac{d}{d u_{2}} [1]$

Step 9.1.3

Evaluate $\frac{d}{d u_{2}} [2 u_{2}]$ .

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Step 9.1.3.1

Since $2$ is constant with respect to $u_{2}$ , the derivative of $2 u_{2}$ with respect to $u_{2}$ is $2 \frac{d}{d u_{2}} [u_{2}]$ .

$2 \frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [1]$

Step 9.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d u_{2}} [1]$

Step 9.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d u_{2}} [1]$

Step 9.1.4

Differentiate using the Constant Rule.

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Step 9.1.4.1

Since $1$ is constant with respect to $u_{2}$ , the derivative of $1$ with respect to $u_{2}$ is $0$ .

$2 + 0$

Step 9.1.4.2

Add $2$ and $0$ .

$2$

Step 9.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\int {(\frac{u_{3}}{2} - \frac{1}{2})}^{2} {u_{3}}^{- \frac{1}{3}} \frac{1}{2} d u_{3}$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and ${(\frac{u_{3}}{2} - \frac{1}{2})}^{2}$ .

$\int \frac{{(\frac{u_{3}}{2} - \frac{1}{2})}^{2}}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 10.2

Combine $\frac{{(\frac{u_{3}}{2} - \frac{1}{2})}^{2}}{2}$ and ${u_{3}}^{- \frac{1}{3}}$ .

$\int \frac{{(\frac{u_{3}}{2} - \frac{1}{2})}^{2} {u_{3}}^{- \frac{1}{3}}}{2} d u_{3}$

Step 10.3

Move ${u_{3}}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{{(\frac{u_{3}}{2} - \frac{1}{2})}^{2}}{2 {u_{3}}^{\frac{1}{3}}} d u_{3}$

Step 11

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{{(\frac{u_{3}}{2} - \frac{1}{2})}^{2}}{{u_{3}}^{\frac{1}{3}}} d u_{3}$

Step 12

Apply basic rules of exponents.

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Step 12.1

Move ${u_{3}}^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} \int {(\frac{u_{3}}{2} - \frac{1}{2})}^{2} {({u_{3}}^{\frac{1}{3}})}^{- 1} d u_{3}$

Step 12.2

Multiply the exponents in ${({u_{3}}^{\frac{1}{3}})}^{- 1}$ .

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Step 12.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \int {(\frac{u_{3}}{2} - \frac{1}{2})}^{2} {u_{3}}^{\frac{1}{3} \cdot - 1} d u_{3}$

Step 12.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{1}{2} \int {(\frac{u_{3}}{2} - \frac{1}{2})}^{2} {u_{3}}^{\frac{- 1}{3}} d u_{3}$

Step 12.2.3

Move the negative in front of the fraction.

$\frac{1}{2} \int {(\frac{u_{3}}{2} - \frac{1}{2})}^{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13

Simplify.

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Step 13.1

Rewrite ${(\frac{u_{3}}{2} - \frac{1}{2})}^{2}$ as $(\frac{u_{3}}{2} - \frac{1}{2}) (\frac{u_{3}}{2} - \frac{1}{2})$ .

$\frac{1}{2} \int (\frac{u_{3}}{2} - \frac{1}{2}) (\frac{u_{3}}{2} - \frac{1}{2}) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.2

Apply the distributive property.

$\frac{1}{2} \int (\frac{u_{3}}{2} (\frac{u_{3}}{2} - \frac{1}{2}) - \frac{1}{2} (\frac{u_{3}}{2} - \frac{1}{2})) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.3

Apply the distributive property.

$\frac{1}{2} \int (\frac{u_{3}}{2} \cdot \frac{u_{3}}{2} + \frac{u_{3}}{2} (- \frac{1}{2}) - \frac{1}{2} (\frac{u_{3}}{2} - \frac{1}{2})) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.4

Apply the distributive property.

$\frac{1}{2} \int (\frac{u_{3}}{2} \cdot \frac{u_{3}}{2} + \frac{u_{3}}{2} (- \frac{1}{2}) - \frac{1}{2} \cdot \frac{u_{3}}{2} - \frac{1}{2} (- \frac{1}{2})) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.5

Apply the distributive property.

$\frac{1}{2} \int (\frac{u_{3}}{2} \cdot \frac{u_{3}}{2} + \frac{u_{3}}{2} (- \frac{1}{2})) {u_{3}}^{- \frac{1}{3}} + (- \frac{1}{2} \cdot \frac{u_{3}}{2} - \frac{1}{2} (- \frac{1}{2})) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.6

Apply the distributive property.

$\frac{1}{2} \int \frac{u_{3}}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} + \frac{u_{3}}{2} (- \frac{1}{2}) {u_{3}}^{- \frac{1}{3}} + (- \frac{1}{2} \cdot \frac{u_{3}}{2} - \frac{1}{2} (- \frac{1}{2})) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.7

Apply the distributive property.

$\frac{1}{2} \int \frac{u_{3}}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} + \frac{u_{3}}{2} (- \frac{1}{2}) {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} (- \frac{1}{2}) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.8

Reorder $\frac{u_{3}}{2}$ and $- 1$ .

$\frac{1}{2} \int \frac{u_{3}}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} (- \frac{1}{2}) {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.9

Move $\frac{1}{2}$ .

$\frac{1}{2} \int \frac{u_{3}}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.10

Multiply $\frac{u_{3}}{2}$ by $\frac{u_{3}}{2}$ .

$\frac{1}{2} \int \frac{u_{3} \cdot u_{3}}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.11

Raise $u_{3}$ to the power of $1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{1} u_{3}}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.12

Raise $u_{3}$ to the power of $1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{1} {u_{3}}^{1}}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.13

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int \frac{{u_{3}}^{1 + 1}}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.14

Add $1$ and $1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{2}}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.15

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{{u_{3}}^{2}}{4} {u_{3}}^{- \frac{1}{3}} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.16

Combine $\frac{{u_{3}}^{2}}{4}$ and ${u_{3}}^{- \frac{1}{3}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{2} {u_{3}}^{- \frac{1}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.17

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int \frac{{u_{3}}^{2 - \frac{1}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.18

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{2 \cdot \frac{3}{3} - \frac{1}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.19

Combine $2$ and $\frac{3}{3}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{2 \cdot 3}{3} - \frac{1}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.20

Combine the numerators over the common denominator.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{2 \cdot 3 - 1}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.21

Simplify the numerator.

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Step 13.21.1

Multiply $2$ by $3$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{6 - 1}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.21.2

Subtract $1$ from $6$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} - 1 \frac{u_{3}}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.22

Combine $- 1 \frac{u_{3}}{2}$ and $\frac{1}{2}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{u_{3}}{2}}{2} {u_{3}}^{- \frac{1}{3}} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.23

Combine $\frac{- 1 \frac{u_{3}}{2}}{2}$ and ${u_{3}}^{- \frac{1}{3}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.24

Combine $\frac{u_{3}}{2}$ and ${u_{3}}^{- \frac{1}{3}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{u_{3} \cdot {u_{3}}^{- \frac{1}{3}}}{2}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.25

Raise $u_{3}$ to the power of $1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{1} {u_{3}}^{- \frac{1}{3}}}{2}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.26

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{1 - \frac{1}{3}}}{2}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.27

Write $1$ as a fraction with a common denominator.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{3}{3} - \frac{1}{3}}}{2}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.28

Combine the numerators over the common denominator.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{3 - 1}{3}}}{2}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.29

Subtract $1$ from $3$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{1}{2} \cdot \frac{u_{3}}{2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.30

Combine $- \frac{1}{2}$ and $\frac{u_{3}}{2}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{u_{3}}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.31

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{u_{3}}{4} {u_{3}}^{- \frac{1}{3}} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.32

Combine $\frac{u_{3}}{4}$ and ${u_{3}}^{- \frac{1}{3}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{u_{3} \cdot {u_{3}}^{- \frac{1}{3}}}{4} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.33

Raise $u_{3}$ to the power of $1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{1} {u_{3}}^{- \frac{1}{3}}}{4} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.34

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{1 - \frac{1}{3}}}{4} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.35

Write $1$ as a fraction with a common denominator.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{3}{3} - \frac{1}{3}}}{4} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.36

Combine the numerators over the common denominator.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{3 - 1}{3}}}{4} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.37

Subtract $1$ from $3$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{2}{3}}}{4} - 1 \cdot - 1 \frac{1}{2} \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.38

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + 1 (\frac{1}{2}) \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.39

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + \frac{1}{2} \cdot \frac{1}{2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.40

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + \frac{1}{2 \cdot 2} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.41

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + \frac{1}{4} {u_{3}}^{- \frac{1}{3}} d u_{3}$

Step 13.42

Combine $\frac{1}{4}$ and ${u_{3}}^{- \frac{1}{3}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + \frac{{u_{3}}^{- \frac{1}{3}}}{4} d u_{3}$

Step 13.43

Reorder $- \frac{{u_{3}}^{\frac{2}{3}}}{4}$ and $\frac{{u_{3}}^{- \frac{1}{3}}}{4}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} + \frac{{u_{3}}^{- \frac{1}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14

Simplify.

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Step 14.1

Rewrite $- 1 \frac{{u_{3}}^{\frac{2}{3}}}{2}$ as $- \frac{{u_{3}}^{\frac{2}{3}}}{2}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{- \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2} + \frac{{u_{3}}^{- \frac{1}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.2

Rewrite $\frac{- \frac{{u_{3}}^{\frac{2}{3}}}{2}}{2}$ as a product.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{2} \cdot \frac{1}{2} + \frac{{u_{3}}^{- \frac{1}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.3

Multiply $\frac{1}{2}$ by $\frac{{u_{3}}^{\frac{2}{3}}}{2}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{2 \cdot 2} + \frac{{u_{3}}^{- \frac{1}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.4

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + \frac{{u_{3}}^{- \frac{1}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.5

Move ${u_{3}}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} - \frac{{u_{3}}^{\frac{2}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} - \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.6

Subtract $\frac{{u_{3}}^{\frac{2}{3}}}{4}$ from $- \frac{{u_{3}}^{\frac{2}{3}}}{4}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} - 2 \frac{{u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.7

Combine $- 2$ and $\frac{{u_{3}}^{\frac{2}{3}}}{4}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} + \frac{- 2 {u_{3}}^{\frac{2}{3}}}{4} d u_{3}$

Step 14.8

Factor $2$ out of $- 2 {u_{3}}^{\frac{2}{3}}$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} + \frac{2 (- {u_{3}}^{\frac{2}{3}})}{4} d u_{3}$

Step 14.9

Cancel the common factors.

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Step 14.9.1

Factor $2$ out of $4$ .

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} + \frac{2 (- {u_{3}}^{\frac{2}{3}})}{2 (2)} d u_{3}$

Step 14.9.2

Cancel the common factor.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} + \frac{2 (- {u_{3}}^{\frac{2}{3}})}{2 \cdot 2} d u_{3}$

Step 14.9.3

Rewrite the expression.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} + \frac{- {u_{3}}^{\frac{2}{3}}}{2} d u_{3}$

Step 14.10

Move the negative in front of the fraction.

$\frac{1}{2} \int \frac{{u_{3}}^{\frac{5}{3}}}{4} + \frac{1}{4 {u_{3}}^{\frac{1}{3}}} - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3}$

Step 15

Split the single integral into multiple integrals.

$\frac{1}{2} (\int \frac{{u_{3}}^{\frac{5}{3}}}{4} d u_{3} + \int \frac{1}{4 {u_{3}}^{\frac{1}{3}}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 16

Since $\frac{1}{4}$ is constant with respect to $u_{3}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} (\frac{1}{4} \int {u_{3}}^{\frac{5}{3}} d u_{3} + \int \frac{1}{4 {u_{3}}^{\frac{1}{3}}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 17

By the Power Rule, the integral of ${u_{3}}^{\frac{5}{3}}$ with respect to $u_{3}$ is $\frac{3}{8} {u_{3}}^{\frac{8}{3}}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3}{8} {u_{3}}^{\frac{8}{3}} + C) + \int \frac{1}{4 {u_{3}}^{\frac{1}{3}}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 18

Combine $\frac{3}{8}$ and ${u_{3}}^{\frac{8}{3}}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \int \frac{1}{4 {u_{3}}^{\frac{1}{3}}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 19

Since $\frac{1}{4}$ is constant with respect to $u_{3}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} \int \frac{1}{{u_{3}}^{\frac{1}{3}}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 20

Apply basic rules of exponents.

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Step 20.1

Move ${u_{3}}^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} \int {({u_{3}}^{\frac{1}{3}})}^{- 1} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 20.2

Multiply the exponents in ${({u_{3}}^{\frac{1}{3}})}^{- 1}$ .

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Step 20.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} \int {u_{3}}^{\frac{1}{3} \cdot - 1} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 20.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} \int {u_{3}}^{\frac{- 1}{3}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 20.2.3

Move the negative in front of the fraction.

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} \int {u_{3}}^{- \frac{1}{3}} d u_{3} + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 21

By the Power Rule, the integral of ${u_{3}}^{- \frac{1}{3}}$ with respect to $u_{3}$ is $\frac{3}{2} {u_{3}}^{\frac{2}{3}}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} (\frac{3}{2} {u_{3}}^{\frac{2}{3}} + C) + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 22

Combine $\frac{3}{2}$ and ${u_{3}}^{\frac{2}{3}}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} (\frac{3 {u_{3}}^{\frac{2}{3}}}{2} + C) + \int - \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 23

Since $- 1$ is constant with respect to $u_{3}$ , move $- 1$ out of the integral.

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} (\frac{3 {u_{3}}^{\frac{2}{3}}}{2} + C) - \int \frac{{u_{3}}^{\frac{2}{3}}}{2} d u_{3})$

Step 24

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} (\frac{3 {u_{3}}^{\frac{2}{3}}}{2} + C) - (\frac{1}{2} \int {u_{3}}^{\frac{2}{3}} d u_{3}))$

Step 25

By the Power Rule, the integral of ${u_{3}}^{\frac{2}{3}}$ with respect to $u_{3}$ is $\frac{3}{5} {u_{3}}^{\frac{5}{3}}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} (\frac{3 {u_{3}}^{\frac{2}{3}}}{2} + C) - \frac{1}{2} (\frac{3}{5} {u_{3}}^{\frac{5}{3}} + C))$

Step 26

Simplify.

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Step 26.1

Combine $\frac{3}{5}$ and ${u_{3}}^{\frac{5}{3}}$ .

$\frac{1}{2} (\frac{1}{4} (\frac{3 {u_{3}}^{\frac{8}{3}}}{8} + C) + \frac{1}{4} (\frac{3 {u_{3}}^{\frac{2}{3}}}{2} + C) - \frac{1}{2} (\frac{3 {u_{3}}^{\frac{5}{3}}}{5} + C))$

Step 26.2

Simplify.

$\frac{1}{2} (\frac{3 {u_{3}}^{\frac{8}{3}}}{32} + \frac{3 {u_{3}}^{\frac{2}{3}}}{8} - \frac{3 {u_{3}}^{\frac{5}{3}}}{10}) + C$

Step 27

Substitute back in for each integration substitution variable.

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Step 27.1

Replace all occurrences of $u_{3}$ with $2 u_{2} + 1$ .

$\frac{1}{2} (\frac{3 {(2 u_{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 u_{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 u_{2} + 1)}^{\frac{5}{3}}}{10}) + C$

Step 27.2

Replace all occurrences of $u_{2}$ with $\frac{u_{1}}{2} - \frac{1}{2}$ .

$\frac{1}{2} (\frac{3 {(2 (\frac{u_{1}}{2} - \frac{1}{2}) + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 (\frac{u_{1}}{2} - \frac{1}{2}) + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{u_{1}}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 27.3

Replace all occurrences of $u_{1}$ with $1 + 2 x$ .

$\frac{1}{2} (\frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28

Simplify.

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Step 28.1

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{3 {(2 \frac{1 + 2 x - 1}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.2

Subtract $1$ from $1$ .

$\frac{1}{2} (\frac{3 {(2 \frac{2 x + 0}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.3

Add $2 x$ and $0$ .

$\frac{1}{2} (\frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.4

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 \frac{1 + 2 x - 1}{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.5

Subtract $1$ from $1$ .

$\frac{1}{2} (\frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 \frac{2 x + 0}{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.6

Add $2 x$ and $0$ .

$\frac{1}{2} (\frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7

Simplify each term.

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Step 28.7.1

Cancel the common factor of $2$ .

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Step 28.7.1.1

Cancel the common factor.

$\frac{1}{2} (\frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.1.2

Divide $x$ by $1$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.2

Cancel the common factor of $2$ .

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Step 28.7.2.1

Cancel the common factor.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.2.2

Divide $x$ by $1$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 (\frac{1 + 2 x}{2} - \frac{1}{2}) + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.3

Simplify each term.

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Step 28.7.3.1

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 \frac{1 + 2 x - 1}{2} + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.3.2

Combine the opposite terms in $1 + 2 x - 1$ .

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Step 28.7.3.2.1

Subtract $1$ from $1$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 \frac{2 x + 0}{2} + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.3.2.2

Add $2 x$ and $0$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.3.3

Cancel the common factor of $2$ .

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Step 28.7.3.3.1

Cancel the common factor.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 \frac{2 x}{2} + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.7.3.3.2

Rewrite the expression.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.8

To write $\frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8} \cdot \frac{4}{4} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.9

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

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Step 28.9.1

Multiply $\frac{3 {(2 x + 1)}^{\frac{2}{3}}}{8}$ by $\frac{4}{4}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}} \cdot 4}{8 \cdot 4} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.9.2

Multiply $8$ by $4$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}}}{32} + \frac{3 {(2 x + 1)}^{\frac{2}{3}} \cdot 4}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.10

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}} + 3 {(2 x + 1)}^{\frac{2}{3}} \cdot 4}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11

Simplify the numerator.

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Step 28.11.1

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 {(2 x + 1)}^{\frac{8}{3}} + 3 {(2 x + 1)}^{\frac{2}{3}} \cdot 4$ .

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Step 28.11.1.1

Move ${(2 x + 1)}^{\frac{2}{3}}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{8}{3}} + 3 \cdot 4 {(2 x + 1)}^{\frac{2}{3}}}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.1.2

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 {(2 x + 1)}^{\frac{8}{3}}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} ({(2 x + 1)}^{\frac{6}{3}}) + 3 \cdot 4 {(2 x + 1)}^{\frac{2}{3}}}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.1.3

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 \cdot 4 {(2 x + 1)}^{\frac{2}{3}}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} ({(2 x + 1)}^{\frac{6}{3}}) + 3 {(2 x + 1)}^{\frac{2}{3}} (4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.1.4

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 {(2 x + 1)}^{\frac{2}{3}} ({(2 x + 1)}^{\frac{6}{3}}) + 3 {(2 x + 1)}^{\frac{2}{3}} (4)$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} ({(2 x + 1)}^{\frac{6}{3}} + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.2

Divide $6$ by $3$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} ({(2 x + 1)}^{2} + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.3

Rewrite ${(2 x + 1)}^{2}$ as $(2 x + 1) (2 x + 1)$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} ((2 x + 1) (2 x + 1) + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.4

Expand $(2 x + 1) (2 x + 1)$ using the FOIL Method.

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Step 28.11.4.1

Apply the distributive property.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (2 x (2 x + 1) + 1 (2 x + 1) + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.4.2

Apply the distributive property.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (2 x (2 x) + 2 x \cdot 1 + 1 (2 x + 1) + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.4.3

Apply the distributive property.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (2 x (2 x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5

Simplify and combine like terms.

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Step 28.11.5.1

Simplify each term.

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Step 28.11.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (2 \cdot 2 x \cdot x + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 28.11.5.1.2.1

Move $x$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (2 \cdot 2 (x \cdot x) + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.1.2.2

Multiply $x$ by $x$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (2 \cdot 2 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.1.3

Multiply $2$ by $2$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.1.4

Multiply $2$ by $1$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 2 x + 1 (2 x) + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.1.5

Multiply $2 x$ by $1$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 2 x + 2 x + 1 \cdot 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.1.6

Multiply $1$ by $1$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 2 x + 2 x + 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.5.2

Add $2 x$ and $2 x$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 1 + 4)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.11.6

Add $1$ and $4$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5)}{32} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.12

To write $\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5)}{32}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5)}{32} \cdot \frac{5}{5} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}) + C$

Step 28.13

To write $- \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5)}{32} \cdot \frac{5}{5} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10} \cdot \frac{16}{16}) + C$

Step 28.14

Write each expression with a common denominator of $160$ , by multiplying each by an appropriate factor of $1$ .

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Step 28.14.1

Multiply $\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5)}{32}$ by $\frac{5}{5}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) \cdot 5}{32 \cdot 5} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10} \cdot \frac{16}{16}) + C$

Step 28.14.2

Multiply $32$ by $5$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) \cdot 5}{160} - \frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10} \cdot \frac{16}{16}) + C$

Step 28.14.3

Multiply $\frac{3 {(2 x + 1)}^{\frac{5}{3}}}{10}$ by $\frac{16}{16}$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) \cdot 5}{160} - \frac{3 {(2 x + 1)}^{\frac{5}{3}} \cdot 16}{10 \cdot 16}) + C$

Step 28.14.4

Multiply $10$ by $16$ .

$\frac{1}{2} (\frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) \cdot 5}{160} - \frac{3 {(2 x + 1)}^{\frac{5}{3}} \cdot 16}{160}) + C$

Step 28.15

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) \cdot 5 - 3 {(2 x + 1)}^{\frac{5}{3}} \cdot 16}{160} + C$

Step 28.16

Simplify the numerator.

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Step 28.16.1

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) \cdot 5 - 3 {(2 x + 1)}^{\frac{5}{3}} \cdot 16$ .

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Step 28.16.1.1

Reorder the expression.

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Step 28.16.1.1.1

Move $4 x^{2} + 4 x + 5$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} \cdot 5 (4 x^{2} + 4 x + 5) - 3 {(2 x + 1)}^{\frac{5}{3}} \cdot 16}{160} + C$

Step 28.16.1.1.2

Move ${(2 x + 1)}^{\frac{2}{3}}$ .

$\frac{1}{2} \cdot \frac{3 \cdot 5 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) - 3 {(2 x + 1)}^{\frac{5}{3}} \cdot 16}{160} + C$

Step 28.16.1.1.3

Move ${(2 x + 1)}^{\frac{5}{3}}$ .

$\frac{1}{2} \cdot \frac{3 \cdot 5 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5) - 3 \cdot 16 {(2 x + 1)}^{\frac{5}{3}}}{160} + C$

Step 28.16.1.2

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 \cdot 5 {(2 x + 1)}^{\frac{2}{3}} (4 x^{2} + 4 x + 5)$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (5 (4 x^{2} + 4 x + 5)) - 3 \cdot 16 {(2 x + 1)}^{\frac{5}{3}}}{160} + C$

Step 28.16.1.3

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $- 3 \cdot 16 {(2 x + 1)}^{\frac{5}{3}}$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (5 (4 x^{2} + 4 x + 5)) + 3 {(2 x + 1)}^{\frac{2}{3}} (- 1 \cdot 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.1.4

Factor $3 {(2 x + 1)}^{\frac{2}{3}}$ out of $3 {(2 x + 1)}^{\frac{2}{3}} (5 (4 x^{2} + 4 x + 5)) + 3 {(2 x + 1)}^{\frac{2}{3}} (- 1 \cdot 16 {(2 x + 1)}^{\frac{3}{3}})$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (5 (4 x^{2} + 4 x + 5) - 1 \cdot 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.2

Multiply $- 1$ by $16$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (5 (4 x^{2} + 4 x + 5) - 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.3

Simplify each term.

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Step 28.16.3.1

Apply the distributive property.

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (5 (4 x^{2}) + 5 (4 x) + 5 \cdot 5 - 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.3.2

Simplify.

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Step 28.16.3.2.1

Multiply $4$ by $5$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 5 (4 x) + 5 \cdot 5 - 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.3.2.2

Multiply $4$ by $5$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 5 \cdot 5 - 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.3.2.3

Multiply $5$ by $5$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 25 - 16 {(2 x + 1)}^{\frac{3}{3}})}{160} + C$

Step 28.16.3.3

Divide $3$ by $3$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 25 - 16 {(2 x + 1)}^{1})}{160} + C$

Step 28.16.3.4

Simplify.

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 25 - 16 (2 x + 1))}{160} + C$

Step 28.16.3.5

Apply the distributive property.

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 25 - 16 (2 x) - 16 \cdot 1)}{160} + C$

Step 28.16.3.6

Multiply $2$ by $- 16$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 25 - 32 x - 16 \cdot 1)}{160} + C$

Step 28.16.3.7

Multiply $- 16$ by $1$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} + 20 x + 25 - 32 x - 16)}{160} + C$

Step 28.16.4

Subtract $32 x$ from $20 x$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} - 12 x + 25 - 16)}{160} + C$

Step 28.16.5

Subtract $16$ from $25$ .

$\frac{1}{2} \cdot \frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} - 12 x + 9)}{160} + C$

Step 28.17

Combine.

$\frac{1 (3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} - 12 x + 9))}{2 \cdot 160} + C$

Step 28.18

Multiply $3$ by $1$ .

$\frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} - 12 x + 9)}{2 \cdot 160} + C$

Step 28.19

Multiply $2$ by $160$ .

$\frac{3 {(2 x + 1)}^{\frac{2}{3}} (20 x^{2} - 12 x + 9)}{320} + C$