Calculus Examples

$\int 5 x^{3} \sqrt{1 - x^{2}} d x$

Step 1

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

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Step 1.1

Let $u = 1 - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 - x^{2}$ .

$\frac{d}{d x} [1 - x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 2$ .

$0 - (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 1.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 1.2

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

$\int 5 {\sqrt{- u + 1}}^{2} \sqrt{u} \frac{1}{- 2} d u$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{- u + 1}}^{2}$ as $- u + 1$ .

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Step 2.1.1

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

$\int 5 {({(- u + 1)}^{\frac{1}{2}})}^{2} \sqrt{u} \frac{1}{- 2} d u$

Step 2.1.2

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

$\int 5 {(- u + 1)}^{\frac{1}{2} \cdot 2} \sqrt{u} \frac{1}{- 2} d u$

Step 2.1.3

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

$\int 5 {(- u + 1)}^{\frac{2}{2}} \sqrt{u} \frac{1}{- 2} d u$

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

$\int 5 {(- u + 1)}^{\frac{2}{2}} \sqrt{u} \frac{1}{- 2} d u$

Step 2.1.4.2

Rewrite the expression.

$\int 5 {(- u + 1)}^{1} \sqrt{u} \frac{1}{- 2} d u$

Step 2.1.5

Simplify.

$\int 5 (- u + 1) \sqrt{u} \frac{1}{- 2} d u$

Step 2.2

Move the negative in front of the fraction.

$\int 5 (- u + 1) \sqrt{u} (- \frac{1}{2}) d u$

Step 2.3

Multiply $- 1$ by $5$ .

$\int - 5 (- u + 1) \sqrt{u} \frac{1}{2} d u$

Step 2.4

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

$\int \frac{- 5}{2} (- u + 1) \sqrt{u} d u$

Step 2.5

Combine $\sqrt{u}$ and $\frac{- 5}{2}$ .

$\int \frac{\sqrt{u} \cdot - 5}{2} (- u + 1) d u$

Step 2.6

Move $- 5$ to the left of $\sqrt{u}$ .

$\int \frac{- 5 \cdot \sqrt{u}}{2} (- u + 1) d u$

Step 2.7

Move the negative in front of the fraction.

$\int - \frac{5 \sqrt{u}}{2} (- u + 1) d u$

Step 3

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

$- \int \frac{5 \sqrt{u}}{2} (- u + 1) d u$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Reorder $\frac{5 u^{\frac{1}{2}}}{2}$ and $- 1$ .

$- \int - 1 \cdot \frac{5 u^{\frac{1}{2}}}{2} u + \frac{5 u^{\frac{1}{2}}}{2} \cdot 1 d u$

Step 5.3

Reorder $\frac{5 u^{\frac{1}{2}}}{2}$ and $1$ .

$- \int - 1 \cdot \frac{5 u^{\frac{1}{2}}}{2} u + 1 \cdot \frac{5 u^{\frac{1}{2}}}{2} d u$

Step 5.4

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

$- \int - 1 \frac{5 u^{\frac{1}{2}} u}{2} + 1 \frac{5 u^{\frac{1}{2}}}{2} d u$

Step 5.5

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

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

Step 5.6

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

$- \int - 1 \frac{5 u^{\frac{1}{2} + 1}}{2} + 1 \frac{5 u^{\frac{1}{2}}}{2} d u$

Step 5.7

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

$- \int - 1 \frac{5 u^{\frac{1}{2} + \frac{2}{2}}}{2} + 1 \frac{5 u^{\frac{1}{2}}}{2} d u$

Step 5.8

Combine the numerators over the common denominator.

$- \int - 1 \frac{5 u^{\frac{1 + 2}{2}}}{2} + 1 \frac{5 u^{\frac{1}{2}}}{2} d u$

Step 5.9

Add $1$ and $2$ .

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

Step 5.10

Multiply $\frac{5 u^{\frac{1}{2}}}{2}$ by $1$ .

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

Step 5.11

Reorder $- 1 \frac{5 u^{\frac{3}{2}}}{2}$ and $\frac{5 u^{\frac{1}{2}}}{2}$ .

$- \int \frac{5 u^{\frac{1}{2}}}{2} - 1 \frac{5 u^{\frac{3}{2}}}{2} d u$

Step 6

Rewrite $- 1 \frac{5 u^{\frac{3}{2}}}{2}$ as $- \frac{5 u^{\frac{3}{2}}}{2}$ .

$- \int \frac{5 u^{\frac{1}{2}}}{2} - \frac{5 u^{\frac{3}{2}}}{2} d u$

Step 7

Split the single integral into multiple integrals.

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

Step 8

Since $\frac{5}{2}$ is constant with respect to $u$ , move $\frac{5}{2}$ out of the integral.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Since $\frac{5}{2}$ is constant with respect to $u$ , move $\frac{5}{2}$ out of the integral.

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

Step 13

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

$- (\frac{5}{2} (\frac{2 u^{\frac{3}{2}}}{3} + C) - \frac{5}{2} (\frac{2}{5} u^{\frac{5}{2}} + C))$

Step 14

Simplify.

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Step 14.1

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

$- (\frac{5}{2} (\frac{2 u^{\frac{3}{2}}}{3} + C) - \frac{5}{2} (\frac{2 u^{\frac{5}{2}}}{5} + C))$

Step 14.2

Simplify.

$- (\frac{5 u^{\frac{3}{2}}}{3} - u^{\frac{5}{2}}) + C$

Step 14.3

Reorder terms.

$- (\frac{5}{3} u^{\frac{3}{2}} - u^{\frac{5}{2}}) + C$

Step 15

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 16

Simplify.

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Step 16.1

Combine $\frac{5}{3}$ and ${(1 - x^{2})}^{\frac{3}{2}}$ .

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

Step 16.2

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

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

Step 16.3

Combine $- {(1 - x^{2})}^{\frac{5}{2}}$ and $\frac{3}{3}$ .

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

Step 16.4

Combine the numerators over the common denominator.

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

Step 16.5

Multiply $3$ by $- 1$ .

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

Step 17

Reorder terms.

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