Calculus Examples

$\frac{3 \sqrt{x} + 4 x^{2}}{x^{4}}$

Step 1

Write $\frac{3 \sqrt{x} + 4 x^{2}}{x^{4}}$ as a function.

$f (x) = \frac{3 \sqrt{x} + 4 x^{2}}{x^{4}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{3 \sqrt{x} + 4 x^{2}}{x^{4}} d x$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int \frac{3 x^{\frac{1}{2}} + 4 x^{2}}{x^{4}} d x$

Step 4.1.2

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

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Step 4.1.2.1

Factor $x^{\frac{1}{2}}$ out of $3 x^{\frac{1}{2}}$ .

$\int \frac{x^{\frac{1}{2}} \cdot 3 + 4 x^{2}}{x^{4}} d x$

Step 4.1.2.2

Factor $x^{\frac{1}{2}}$ out of $4 x^{2}$ .

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

Step 4.1.2.3

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

$\int \frac{x^{\frac{1}{2}} (3 + 4 x^{\frac{3}{2}})}{x^{4}} d x$

Step 4.2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 4.3

Multiply $x^{4}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.3.1

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

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

Step 4.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3.3

Combine $4$ and $\frac{2}{2}$ .

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

Step 4.3.4

Combine the numerators over the common denominator.

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

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $4$ by $2$ .

$\int \frac{3 + 4 x^{\frac{3}{2}}}{x^{\frac{8 - 1}{2}}} d x$

Step 4.3.5.2

Subtract $1$ from $8$ .

$\int \frac{3 + 4 x^{\frac{3}{2}}}{x^{\frac{7}{2}}} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $x^{\frac{7}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int (3 + 4 x^{\frac{3}{2}}) {(x^{\frac{7}{2}})}^{- 1} d x$

Step 5.2

Multiply the exponents in ${(x^{\frac{7}{2}})}^{- 1}$ .

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Step 5.2.1

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

$\int (3 + 4 x^{\frac{3}{2}}) x^{\frac{7}{2} \cdot - 1} d x$

Step 5.2.2

Multiply $\frac{7}{2} \cdot - 1$ .

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Step 5.2.2.1

Combine $\frac{7}{2}$ and $- 1$ .

$\int (3 + 4 x^{\frac{3}{2}}) x^{\frac{7 \cdot - 1}{2}} d x$

Step 5.2.2.2

Multiply $7$ by $- 1$ .

$\int (3 + 4 x^{\frac{3}{2}}) x^{\frac{- 7}{2}} d x$

Step 5.2.3

Move the negative in front of the fraction.

$\int (3 + 4 x^{\frac{3}{2}}) x^{- \frac{7}{2}} d x$

Step 6

Let $u_{1} = 3 + 4 x^{\frac{3}{2}}$ . Then $d u_{1} = 6 x^{\frac{1}{2}} d x$ , so $\frac{1}{6} d u_{1} = x^{\frac{1}{2}} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 6.1

Let $u_{1} = 3 + 4 x^{\frac{3}{2}}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.1.1

Differentiate $3 + 4 x^{\frac{3}{2}}$ .

$\frac{d}{d x} [3 + 4 x^{\frac{3}{2}}]$

Step 6.1.2

Differentiate.

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Step 6.1.2.1

By the Sum Rule, the derivative of $3 + 4 x^{\frac{3}{2}}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [4 x^{\frac{3}{2}}]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [4 x^{\frac{3}{2}}]$

Step 6.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 x^{\frac{3}{2}}]$

Step 6.1.3

Evaluate $\frac{d}{d x} [4 x^{\frac{3}{2}}]$ .

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Step 6.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{3}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$0 + 4 \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 6.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$0 + 4 (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 6.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0 + 4 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 6.1.3.4

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

$0 + 4 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6.1.3.5

Combine the numerators over the common denominator.

$0 + 4 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 6.1.3.6

Simplify the numerator.

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Step 6.1.3.6.1

Multiply $- 1$ by $2$ .

$0 + 4 (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 6.1.3.6.2

Subtract $2$ from $3$ .

$0 + 4 (\frac{3}{2} x^{\frac{1}{2}})$

Step 6.1.3.7

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

$0 + 4 \frac{3 x^{\frac{1}{2}}}{2}$

Step 6.1.3.8

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

$0 + \frac{4 (3 x^{\frac{1}{2}})}{2}$

Step 6.1.3.9

Multiply $3$ by $4$ .

$0 + \frac{12 x^{\frac{1}{2}}}{2}$

Step 6.1.3.10

Factor $2$ out of $12 x^{\frac{1}{2}}$ .

$0 + \frac{2 (6 x^{\frac{1}{2}})}{2}$

Step 6.1.3.11

Cancel the common factors.

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Step 6.1.3.11.1

Factor $2$ out of $2$ .

$0 + \frac{2 (6 x^{\frac{1}{2}})}{2 (1)}$

Step 6.1.3.11.2

Cancel the common factor.

$0 + \frac{2 (6 x^{\frac{1}{2}})}{2 \cdot 1}$

Step 6.1.3.11.3

Rewrite the expression.

$0 + \frac{6 x^{\frac{1}{2}}}{1}$

Step 6.1.3.11.4

Divide $6 x^{\frac{1}{2}}$ by $1$ .

$0 + 6 x^{\frac{1}{2}}$

Step 6.1.4

Add $0$ and $6 x^{\frac{1}{2}}$ .

$6 x^{\frac{1}{2}}$

Step 6.2

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

$\int u_{1} {(\frac{{(- 3 + u_{1})}^{\frac{2}{3}}}{4^{\frac{2}{3}}})}^{\frac{- 8}{2}} \frac{1}{6} d u_{1}$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Cancel the common factor of $- 8$ and $2$ .

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Step 7.1.1.1

Factor $2$ out of $- 8$ .

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

Step 7.1.1.2

Cancel the common factors.

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Step 7.1.1.2.1

Factor $2$ out of $2$ .

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

Step 7.1.1.2.2

Cancel the common factor.

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

Step 7.1.1.2.3

Rewrite the expression.

$\int u_{1} {(\frac{{(- 3 + u_{1})}^{\frac{2}{3}}}{4^{\frac{2}{3}}})}^{\frac{- 4}{1}} \frac{1}{6} d u_{1}$

Step 7.1.1.2.4

Divide $- 4$ by $1$ .

$\int u_{1} {(\frac{{(- 3 + u_{1})}^{\frac{2}{3}}}{4^{\frac{2}{3}}})}^{- 4} \frac{1}{6} d u_{1}$

Step 7.1.2

Combine $\frac{1}{6}$ and $u_{1}$ .

$\int \frac{u_{1}}{6} {(\frac{{(- 3 + u_{1})}^{\frac{2}{3}}}{4^{\frac{2}{3}}})}^{- 4} d u_{1}$

Step 7.2

Apply the product rule to $\frac{{(- 3 + u_{1})}^{\frac{2}{3}}}{4^{\frac{2}{3}}}$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Multiply the exponents in ${({(- 3 + u_{1})}^{\frac{2}{3}})}^{- 4}$ .

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Step 7.3.1.1

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

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

Step 7.3.1.2

Multiply $\frac{2}{3} \cdot - 4$ .

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Step 7.3.1.2.1

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

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

Step 7.3.1.2.2

Multiply $2$ by $- 4$ .

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{\frac{- 8}{3}}}{{(4^{\frac{2}{3}})}^{- 4}} d u_{1}$

Step 7.3.1.3

Move the negative in front of the fraction.

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{- \frac{8}{3}}}{{(4^{\frac{2}{3}})}^{- 4}} d u_{1}$

Step 7.3.2

Multiply the exponents in ${(4^{\frac{2}{3}})}^{- 4}$ .

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Step 7.3.2.1

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

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{- \frac{8}{3}}}{4^{\frac{2}{3} \cdot - 4}} d u_{1}$

Step 7.3.2.2

Multiply $\frac{2}{3} \cdot - 4$ .

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Step 7.3.2.2.1

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

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{- \frac{8}{3}}}{4^{\frac{2 \cdot - 4}{3}}} d u_{1}$

Step 7.3.2.2.2

Multiply $2$ by $- 4$ .

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{- \frac{8}{3}}}{4^{\frac{- 8}{3}}} d u_{1}$

Step 7.3.2.3

Move the negative in front of the fraction.

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{- \frac{8}{3}}}{4^{- \frac{8}{3}}} d u_{1}$

Step 7.3.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{u_{1}}{6} \cdot \frac{{(- 3 + u_{1})}^{- \frac{8}{3}}}{\frac{1}{4^{\frac{8}{3}}}} d u_{1}$

Step 7.3.4

Move ${(- 3 + u_{1})}^{- \frac{8}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{u_{1}}{6} \cdot \frac{1}{\frac{1}{4^{\frac{8}{3}}} {(- 3 + u_{1})}^{\frac{8}{3}}} d u_{1}$

Step 7.3.5

Combine $\frac{1}{4^{\frac{8}{3}}}$ and ${(- 3 + u_{1})}^{\frac{8}{3}}$ .

$\int \frac{u_{1}}{6} \cdot \frac{1}{\frac{{(- 3 + u_{1})}^{\frac{8}{3}}}{4^{\frac{8}{3}}}} d u_{1}$

Step 7.3.6

Multiply by the reciprocal of the fraction to divide by $\frac{{(- 3 + u_{1})}^{\frac{8}{3}}}{4^{\frac{8}{3}}}$ .

$\int \frac{u_{1}}{6} (1 \frac{4^{\frac{8}{3}}}{{(- 3 + u_{1})}^{\frac{8}{3}}}) d u_{1}$

Step 7.3.7

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

$\int \frac{u_{1}}{6} \cdot \frac{4^{\frac{8}{3}}}{{(- 3 + u_{1})}^{\frac{8}{3}}} d u_{1}$

Step 7.3.8

Multiply $\frac{u_{1}}{6}$ by $\frac{4^{\frac{8}{3}}}{{(- 3 + u_{1})}^{\frac{8}{3}}}$ .

$\int \frac{u_{1} \cdot 4^{\frac{8}{3}}}{6 {(- 3 + u_{1})}^{\frac{8}{3}}} d u_{1}$

Step 8

Since $\frac{4^{\frac{8}{3}}}{6}$ is constant with respect to $u_{1}$ , move $\frac{4^{\frac{8}{3}}}{6}$ out of the integral.

$\frac{4^{\frac{8}{3}}}{6} \int \frac{u_{1}}{{(- 3 + u_{1})}^{\frac{8}{3}}} d u_{1}$

Step 9

Let $u_{2} = - 3 + u_{1}$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 9.1

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

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Step 9.1.1

Differentiate $- 3 + u_{1}$ .

$\frac{d}{d u_{1}} [- 3 + u_{1}]$

Step 9.1.2

By the Sum Rule, the derivative of $- 3 + u_{1}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [- 3] + \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{d}{d u_{1}} [- 3] + \frac{d}{d u_{1}} [u_{1}]$

Step 9.1.3

Since $- 3$ is constant with respect to $u_{1}$ , the derivative of $- 3$ with respect to $u_{1}$ is $0$ .

$0 + \frac{d}{d u_{1}} [u_{1}]$

Step 9.1.4

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

$0 + 1$

Step 9.1.5

Add $0$ and $1$ .

$1$

Step 9.2

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

$\frac{4^{\frac{8}{3}}}{6} \int \frac{u_{2} + 3}{{u_{2}}^{\frac{8}{3}}} d u_{2}$

Step 10

Apply basic rules of exponents.

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Step 10.1

Move ${u_{2}}^{\frac{8}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{4^{\frac{8}{3}}}{6} \int (u_{2} + 3) {({u_{2}}^{\frac{8}{3}})}^{- 1} d u_{2}$

Step 10.2

Multiply the exponents in ${({u_{2}}^{\frac{8}{3}})}^{- 1}$ .

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Step 10.2.1

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

$\frac{4^{\frac{8}{3}}}{6} \int (u_{2} + 3) {u_{2}}^{\frac{8}{3} \cdot - 1} d u_{2}$

Step 10.2.2

Multiply $\frac{8}{3} \cdot - 1$ .

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Step 10.2.2.1

Combine $\frac{8}{3}$ and $- 1$ .

$\frac{4^{\frac{8}{3}}}{6} \int (u_{2} + 3) {u_{2}}^{\frac{8 \cdot - 1}{3}} d u_{2}$

Step 10.2.2.2

Multiply $8$ by $- 1$ .

$\frac{4^{\frac{8}{3}}}{6} \int (u_{2} + 3) {u_{2}}^{\frac{- 8}{3}} d u_{2}$

Step 10.2.3

Move the negative in front of the fraction.

$\frac{4^{\frac{8}{3}}}{6} \int (u_{2} + 3) {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11

Expand $(u_{2} + 3) {u_{2}}^{- \frac{8}{3}}$ .

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Step 11.1

Apply the distributive property.

$\frac{4^{\frac{8}{3}}}{6} \int u_{2} \cdot {u_{2}}^{- \frac{8}{3}} + 3 {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11.2

Raise $u_{2}$ to the power of $1$ .

$\frac{4^{\frac{8}{3}}}{6} \int {u_{2}}^{1} {u_{2}}^{- \frac{8}{3}} + 3 {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11.3

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

$\frac{4^{\frac{8}{3}}}{6} \int {u_{2}}^{1 - \frac{8}{3}} + 3 {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11.4

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

$\frac{4^{\frac{8}{3}}}{6} \int {u_{2}}^{\frac{3}{3} - \frac{8}{3}} + 3 {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11.5

Combine the numerators over the common denominator.

$\frac{4^{\frac{8}{3}}}{6} \int {u_{2}}^{\frac{3 - 8}{3}} + 3 {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11.6

Subtract $8$ from $3$ .

$\frac{4^{\frac{8}{3}}}{6} \int {u_{2}}^{\frac{- 5}{3}} + 3 {u_{2}}^{- \frac{8}{3}} d u_{2}$

Step 11.7

Reorder ${u_{2}}^{\frac{- 5}{3}}$ and $3 {u_{2}}^{- \frac{8}{3}}$ .

$\frac{4^{\frac{8}{3}}}{6} \int 3 {u_{2}}^{- \frac{8}{3}} + {u_{2}}^{\frac{- 5}{3}} d u_{2}$

Step 12

Move the negative in front of the fraction.

$\frac{4^{\frac{8}{3}}}{6} \int 3 {u_{2}}^{- \frac{8}{3}} + {u_{2}}^{- \frac{5}{3}} d u_{2}$

Step 13

Split the single integral into multiple integrals.

$\frac{4^{\frac{8}{3}}}{6} (\int 3 {u_{2}}^{- \frac{8}{3}} d u_{2} + \int {u_{2}}^{- \frac{5}{3}} d u_{2})$

Step 14

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

$\frac{4^{\frac{8}{3}}}{6} (3 \int {u_{2}}^{- \frac{8}{3}} d u_{2} + \int {u_{2}}^{- \frac{5}{3}} d u_{2})$

Step 15

By the Power Rule, the integral of ${u_{2}}^{- \frac{8}{3}}$ with respect to $u_{2}$ is $- \frac{3}{5} {u_{2}}^{- \frac{5}{3}}$ .

$\frac{4^{\frac{8}{3}}}{6} (3 (- \frac{3}{5} {u_{2}}^{- \frac{5}{3}} + C) + \int {u_{2}}^{- \frac{5}{3}} d u_{2})$

Step 16

Simplify.

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Step 16.1

Combine ${u_{2}}^{- \frac{5}{3}}$ and $\frac{3}{5}$ .

$\frac{4^{\frac{8}{3}}}{6} (3 (- \frac{{u_{2}}^{- \frac{5}{3}} \cdot 3}{5} + C) + \int {u_{2}}^{- \frac{5}{3}} d u_{2})$

Step 16.2

Move $3$ to the left of ${u_{2}}^{- \frac{5}{3}}$ .

$\frac{4^{\frac{8}{3}}}{6} (3 (- \frac{3 \cdot {u_{2}}^{- \frac{5}{3}}}{5} + C) + \int {u_{2}}^{- \frac{5}{3}} d u_{2})$

Step 16.3

Move ${u_{2}}^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4^{\frac{8}{3}}}{6} (3 (- \frac{3}{5 {u_{2}}^{\frac{5}{3}}} + C) + \int {u_{2}}^{- \frac{5}{3}} d u_{2})$

Step 17

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

$\frac{4^{\frac{8}{3}}}{6} (3 (- \frac{3}{5 {u_{2}}^{\frac{5}{3}}} + C) - \frac{3}{2} {u_{2}}^{- \frac{2}{3}} + C)$

Step 18

Simplify.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {u_{2}}^{\frac{5}{3}}} - \frac{3}{2 {u_{2}}^{\frac{2}{3}}}) + C$

Step 19

Substitute back in for each integration substitution variable.

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Step 19.1

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {(- 3 + u_{1})}^{\frac{5}{3}}} - \frac{3}{2 {(- 3 + u_{1})}^{\frac{2}{3}}}) + C$

Step 19.2

Replace all occurrences of $u_{1}$ with $3 + 4 x^{\frac{3}{2}}$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {(- 3 + 3 + 4 x^{\frac{3}{2}})}^{\frac{5}{3}}} - \frac{3}{2 {(- 3 + 3 + 4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20

Simplify.

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Step 20.1

Combine the opposite terms in $- \frac{9}{5 {(- 3 + 3 + 4 x^{\frac{3}{2}})}^{\frac{5}{3}}} - \frac{3}{2 {(- 3 + 3 + 4 x^{\frac{3}{2}})}^{\frac{2}{3}}}$ .

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Step 20.1.1

Add $- 3$ and $3$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {(0 + 4 x^{\frac{3}{2}})}^{\frac{5}{3}}} - \frac{3}{2 {(- 3 + 3 + 4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.1.2

Add $0$ and $4 x^{\frac{3}{2}}$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {(4 x^{\frac{3}{2}})}^{\frac{5}{3}}} - \frac{3}{2 {(- 3 + 3 + 4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.1.3

Add $- 3$ and $3$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {(4 x^{\frac{3}{2}})}^{\frac{5}{3}}} - \frac{3}{2 {(0 + 4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.1.4

Add $0$ and $4 x^{\frac{3}{2}}$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 {(4 x^{\frac{3}{2}})}^{\frac{5}{3}}} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.2

Simplify each term.

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Step 20.2.1

Simplify the denominator.

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Step 20.2.1.1

Apply the product rule to $4 x^{\frac{3}{2}}$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 (4^{\frac{5}{3}} {(x^{\frac{3}{2}})}^{\frac{5}{3}})} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.2.1.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{5}{3}}$ .

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Step 20.2.1.2.1

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 (4^{\frac{5}{3}} x^{\frac{3}{2} \cdot \frac{5}{3}})} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.2.1.2.2

Cancel the common factor of $3$ .

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Step 20.2.1.2.2.1

Cancel the common factor.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 (4^{\frac{5}{3}} x^{\frac{3}{2} \cdot \frac{5}{3}})} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.2.1.2.2.2

Rewrite the expression.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 (4^{\frac{5}{3}} x^{\frac{1}{2} \cdot 5})} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.2.1.2.3

Combine $\frac{1}{2}$ and $5$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 (4^{\frac{5}{3}} x^{\frac{5}{2}})} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 {(4 x^{\frac{3}{2}})}^{\frac{2}{3}}}) + C$

Step 20.2.2

Simplify the denominator.

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Step 20.2.2.1

Apply the product rule to $4 x^{\frac{3}{2}}$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 (4^{\frac{2}{3}} {(x^{\frac{3}{2}})}^{\frac{2}{3}})}) + C$

Step 20.2.2.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 20.2.2.2.1

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 (4^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}})}) + C$

Step 20.2.2.2.2

Cancel the common factor of $3$ .

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Step 20.2.2.2.2.1

Cancel the common factor.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 (4^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}})}) + C$

Step 20.2.2.2.2.2

Rewrite the expression.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 (4^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2})}) + C$

Step 20.2.2.2.3

Cancel the common factor of $2$ .

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Step 20.2.2.2.3.1

Cancel the common factor.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 (4^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2})}) + C$

Step 20.2.2.2.3.2

Rewrite the expression.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 (4^{\frac{2}{3}} x^{1})}) + C$

Step 20.2.2.3

Simplify.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 \cdot 4^{\frac{2}{3}} x}) + C$

Step 20.2.2.4

Combine exponents.

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Step 20.2.2.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 \cdot {(2^{2})}^{\frac{2}{3}} x}) + C$

Step 20.2.2.4.2

Multiply the exponents in ${(2^{2})}^{\frac{2}{3}}$ .

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Step 20.2.2.4.2.1

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 \cdot 2^{2 (\frac{2}{3})} x}) + C$

Step 20.2.2.4.2.2

Multiply $2 (\frac{2}{3})$ .

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Step 20.2.2.4.2.2.1

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 \cdot 2^{\frac{2 \cdot 2}{3}} x}) + C$

Step 20.2.2.4.2.2.2

Multiply $2$ by $2$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2 \cdot 2^{\frac{4}{3}} x}) + C$

Step 20.2.2.4.3

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2^{1 + \frac{4}{3}} x}) + C$

Step 20.2.2.4.4

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

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2^{\frac{3}{3} + \frac{4}{3}} x}) + C$

Step 20.2.2.4.5

Combine the numerators over the common denominator.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2^{\frac{3 + 4}{3}} x}) + C$

Step 20.2.2.4.6

Add $3$ and $4$ .

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} - \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.3

Apply the distributive property.

$\frac{4^{\frac{8}{3}}}{6} (- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}}) + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.4

Cancel the common factor of $4^{\frac{5}{3}}$ .

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Step 20.4.1

Move the leading negative in $- \frac{9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}}$ into the numerator.

$\frac{4^{\frac{8}{3}}}{6} \cdot \frac{- 9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.4.2

Factor $4^{\frac{5}{3}}$ out of $4^{\frac{8}{3}}$ .

$\frac{4^{\frac{5}{3}} \cdot 4^{\frac{3}{3}}}{6} \cdot \frac{- 9}{5 \cdot 4^{\frac{5}{3}} x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.4.3

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

$\frac{4^{\frac{5}{3}} \cdot 4^{\frac{3}{3}}}{6} \cdot \frac{- 9}{4^{\frac{5}{3}} (5 x^{\frac{5}{2}})} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.4.4

Cancel the common factor.

$\frac{4^{\frac{5}{3}} \cdot 4^{\frac{3}{3}}}{6} \cdot \frac{- 9}{4^{\frac{5}{3}} (5 x^{\frac{5}{2}})} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.4.5

Rewrite the expression.

$\frac{4^{\frac{3}{3}}}{6} \cdot \frac{- 9}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.5

Cancel the common factor of $3$ .

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Step 20.5.1

Factor $3$ out of $6$ .

$\frac{4^{\frac{3}{3}}}{3 (2)} \cdot \frac{- 9}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.5.2

Factor $3$ out of $- 9$ .

$\frac{4^{\frac{3}{3}}}{3 \cdot 2} \cdot \frac{3 \cdot - 3}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.5.3

Cancel the common factor.

$\frac{4^{\frac{3}{3}}}{3 \cdot 2} \cdot \frac{3 \cdot - 3}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.5.4

Rewrite the expression.

$\frac{4^{\frac{3}{3}}}{2} \cdot \frac{- 3}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.6

Multiply $\frac{4^{\frac{3}{3}}}{2}$ by $\frac{- 3}{5 x^{\frac{5}{2}}}$ .

$\frac{4^{\frac{3}{3}} \cdot - 3}{2 (5 x^{\frac{5}{2}})} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.7

Multiply $5$ by $2$ .

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} (- \frac{3}{2^{\frac{7}{3}} x}) + C$

Step 20.8

Cancel the common factor of $3$ .

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Step 20.8.1

Move the leading negative in $- \frac{3}{2^{\frac{7}{3}} x}$ into the numerator.

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{6} \cdot \frac{- 3}{2^{\frac{7}{3}} x} + C$

Step 20.8.2

Factor $3$ out of $6$ .

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{3 (2)} \cdot \frac{- 3}{2^{\frac{7}{3}} x} + C$

Step 20.8.3

Factor $3$ out of $- 3$ .

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{3 \cdot 2} \cdot \frac{3 \cdot - 1}{2^{\frac{7}{3}} x} + C$

Step 20.8.4

Cancel the common factor.

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{3 \cdot 2} \cdot \frac{3 \cdot - 1}{2^{\frac{7}{3}} x} + C$

Step 20.8.5

Rewrite the expression.

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}}}{2} \cdot \frac{- 1}{2^{\frac{7}{3}} x} + C$

Step 20.9

Multiply $\frac{4^{\frac{8}{3}}}{2}$ by $\frac{- 1}{2^{\frac{7}{3}} x}$ .

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2 (2^{\frac{7}{3}} x)} + C$

Step 20.10

Raise $2$ to the power of $1$ .

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{7}{3}} \cdot 2^{1} x} + C$

Step 20.11

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

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{7}{3} + 1} x} + C$

Step 20.12

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

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{7}{3} + \frac{3}{3}} x} + C$

Step 20.13

Combine the numerators over the common denominator.

$\frac{4^{\frac{3}{3}} \cdot - 3}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{7 + 3}{3}} x} + C$

Step 20.14

Add $7$ and $3$ .

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

Step 20.15

Simplify each term.

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Step 20.15.1

Cancel the common factor of $3$ .

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Step 20.15.1.1

Cancel the common factor.

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

Step 20.15.1.2

Rewrite the expression.

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

Step 20.15.2

Evaluate the exponent.

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

Step 20.15.3

Multiply $4$ by $- 3$ .

$\frac{- 12}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{10}{3}} x} + C$

Step 20.15.4

Factor $2$ out of $- 12$ .

$\frac{2 (- 6)}{10 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{10}{3}} x} + C$

Step 20.15.5

Cancel the common factors.

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Step 20.15.5.1

Factor $2$ out of $10 x^{\frac{5}{2}}$ .

$\frac{2 (- 6)}{2 (5 x^{\frac{5}{2}})} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{10}{3}} x} + C$

Step 20.15.5.2

Cancel the common factor.

$\frac{2 \cdot - 6}{2 (5 x^{\frac{5}{2}})} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{10}{3}} x} + C$

Step 20.15.5.3

Rewrite the expression.

$\frac{- 6}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{10}{3}} x} + C$

Step 20.15.6

Move the negative in front of the fraction.

$- \frac{6}{5 x^{\frac{5}{2}}} + \frac{4^{\frac{8}{3}} \cdot - 1}{2^{\frac{10}{3}} x} + C$

Step 20.15.7

Simplify the numerator.

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Step 20.15.7.1

Move $- 1$ to the left of $4^{\frac{8}{3}}$ .

$- \frac{6}{5 x^{\frac{5}{2}}} + \frac{- 1 \cdot 4^{\frac{8}{3}}}{2^{\frac{10}{3}} x} + C$

Step 20.15.7.2

Rewrite $- 1 \cdot 4^{\frac{8}{3}}$ as $- 4^{\frac{8}{3}}$ .

$- \frac{6}{5 x^{\frac{5}{2}}} + \frac{- 4^{\frac{8}{3}}}{2^{\frac{10}{3}} x} + C$

Step 20.15.8

Move the negative in front of the fraction.

$- \frac{6}{5 x^{\frac{5}{2}}} - \frac{4^{\frac{8}{3}}}{2^{\frac{10}{3}} x} + C$

Step 20.16

Reorder factors in $- \frac{6}{5 x^{\frac{5}{2}}} - \frac{4^{\frac{8}{3}}}{2^{\frac{10}{3}} x}$ .

$- \frac{6}{5 x^{\frac{5}{2}}} - \frac{4^{\frac{8}{3}}}{x \cdot 2^{\frac{10}{3}}} + C$

Step 21

The answer is the antiderivative of the function $f (x) = \frac{3 \sqrt{x} + 4 x^{2}}{x^{4}}$ .

$F (x) =$ $- \frac{6}{5 x^{\frac{5}{2}}} - \frac{4^{\frac{8}{3}}}{x \cdot 2^{\frac{10}{3}}} + C$