Calculus Examples

$\int \sqrt{3 - 2 x} x^{2} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \sqrt{3 - 2 x}$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

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

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

Step 4.3

Move the negative in front of the fraction.

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

Step 4.4

Multiply $- 1$ by $- 1$ .

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

Step 4.5

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

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

Step 5

Let $u = 3 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 3 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $3 - 2 x$ .

$\frac{d}{d x} [3 - 2 x]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

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 5.1.3.2

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 5.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 5.1.4

Subtract $2$ from $0$ .

$- 2$

Step 5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

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

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

Step 7

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Combine the numerators over the common denominator.

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

Step 8.7

Add $2$ and $3$ .

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- \int - \frac{u^{\frac{5}{2}}}{2 \cdot 2} + \frac{3}{2} \cdot \frac{u^{\frac{3}{2}}}{2} d u)$

Step 8.8

Multiply $2$ by $2$ .

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

Step 8.9

Multiply $\frac{3}{2}$ by $\frac{u^{\frac{3}{2}}}{2}$ .

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

Step 8.10

Multiply $2$ by $2$ .

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

Step 8.11

Reorder $- \frac{u^{\frac{5}{2}}}{4}$ and $\frac{3 u^{\frac{3}{2}}}{4}$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

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

Step 11

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

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- (\frac{3}{4} (\frac{2}{5} u^{\frac{5}{2}} + C) + \int - \frac{u^{\frac{5}{2}}}{4} d u))$

Step 12

Combine $\frac{2}{5}$ and $u^{\frac{5}{2}}$ .

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- (\frac{3}{4} (\frac{2 u^{\frac{5}{2}}}{5} + C) + \int - \frac{u^{\frac{5}{2}}}{4} d u))$

Step 13

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- (\frac{3}{4} (\frac{2 u^{\frac{5}{2}}}{5} + C) - \int \frac{u^{\frac{5}{2}}}{4} d u))$

Step 14

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

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

Step 15

By the Power Rule, the integral of $u^{\frac{5}{2}}$ with respect to $u$ is $\frac{2}{7} u^{\frac{7}{2}}$ .

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

Step 16

Simplify.

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Step 16.1

Combine $\frac{2}{7}$ and $u^{\frac{7}{2}}$ .

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

Step 16.2

Simplify.

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14})) + C$

Step 16.3

Simplify.

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Step 16.3.1

To write $\frac{2}{3} (- (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14})) \cdot \frac{3}{3} + C$

Step 16.3.2

Combine $\frac{2}{3} (- (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}))$ and $\frac{3}{3}$ .

$- \frac{x^{2} {(3 - 2 x)}^{\frac{3}{2}}}{3} + \frac{\frac{2}{3} (- (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14})) \cdot 3}{3} + C$

Step 16.3.3

Combine the numerators over the common denominator.

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} + \frac{2}{3} (- (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14})) \cdot 3}{3} + C$

Step 16.3.4

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

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot \frac{2 \cdot 3}{3}}{3} + C$

Step 16.3.5

Multiply $2$ by $3$ .

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot \frac{6}{3}}{3} + C$

Step 16.3.6

Cancel the common factor of $6$ and $3$ .

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Step 16.3.6.1

Factor $3$ out of $6$ .

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot \frac{3 \cdot 2}{3}}{3} + C$

Step 16.3.6.2

Cancel the common factors.

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Step 16.3.6.2.1

Factor $3$ out of $3$ .

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot \frac{3 \cdot 2}{3 (1)}}{3} + C$

Step 16.3.6.2.2

Cancel the common factor.

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot \frac{3 \cdot 2}{3 \cdot 1}}{3} + C$

Step 16.3.6.2.3

Rewrite the expression.

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot \frac{2}{1}}{3} + C$

Step 16.3.6.2.4

Divide $2$ by $1$ .

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14}) \cdot 2}{3} + C$

Step 16.3.7

Multiply $2$ by $- 1$ .

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - 2 (\frac{3 u^{\frac{5}{2}}}{10} - \frac{u^{\frac{7}{2}}}{14})}{3} + C$

Step 17

Replace all occurrences of $u$ with $3 - 2 x$ .

$\frac{- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - 2 (\frac{3 {(3 - 2 x)}^{\frac{5}{2}}}{10} - \frac{{(3 - 2 x)}^{\frac{7}{2}}}{14})}{3} + C$

Step 18

Reorder terms.

$\frac{1}{3} (- x^{2} {(3 - 2 x)}^{\frac{3}{2}} - 2 (\frac{3}{10} {(3 - 2 x)}^{\frac{5}{2}} - \frac{1}{14} {(3 - 2 x)}^{\frac{7}{2}})) + C$