Calculus Examples

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

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 $d u$ .

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

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 ${(- u + 1)}^{\frac{1}{2}}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $2$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

Simplify.

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

Step 2.2

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

Factor out negative.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

Combine the numerators over the common denominator.

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

Step 5.8

Add $2$ and $1$ .

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

Step 5.9

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

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

Step 6

Move the negative in front of the fraction.

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Simplify.

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

Step 15

Reorder terms.

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

Step 16

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

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

Step 17

Simplify.

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Step 17.1

Simplify each term.

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Step 17.1.1

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

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

Step 17.1.2

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

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

Step 17.2

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

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

Step 17.3

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

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

Step 17.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.4.1

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

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

Step 17.4.2

Multiply $5$ by $3$ .

$- (- \frac{{(1 - x^{2})}^{\frac{5}{2}} \cdot 3}{15} + \frac{{(1 - x^{2})}^{\frac{3}{2}}}{3} \cdot \frac{5}{5}) + C$

Step 17.4.3

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

$- (- \frac{{(1 - x^{2})}^{\frac{5}{2}} \cdot 3}{15} + \frac{{(1 - x^{2})}^{\frac{3}{2}} \cdot 5}{3 \cdot 5}) + C$

Step 17.4.4

Multiply $3$ by $5$ .

$- (- \frac{{(1 - x^{2})}^{\frac{5}{2}} \cdot 3}{15} + \frac{{(1 - x^{2})}^{\frac{3}{2}} \cdot 5}{15}) + C$

Step 17.5

Combine the numerators over the common denominator.

$- \frac{- {(1 - x^{2})}^{\frac{5}{2}} \cdot 3 + {(1 - x^{2})}^{\frac{3}{2}} \cdot 5}{15} + C$

Step 17.6

Simplify the numerator.

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Step 17.6.1

Multiply $3$ by $- 1$ .

$- \frac{- 3 {(1 - x^{2})}^{\frac{5}{2}} + {(1 - x^{2})}^{\frac{3}{2}} \cdot 5}{15} + C$

Step 17.6.2

Move $5$ to the left of ${(1 - x^{2})}^{\frac{3}{2}}$ .

$- \frac{- 3 {(1 - x^{2})}^{\frac{5}{2}} + 5 {(1 - x^{2})}^{\frac{3}{2}}}{15} + C$

Step 17.7

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

$- \frac{- (3 {(1 - x^{2})}^{\frac{5}{2}}) + 5 {(1 - x^{2})}^{\frac{3}{2}}}{15} + C$

Step 17.8

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

$- \frac{- (3 {(1 - x^{2})}^{\frac{5}{2}}) - (- 5 {(1 - x^{2})}^{\frac{3}{2}})}{15} + C$

Step 17.9

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

$- \frac{- (3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}})}{15} + C$

Step 17.10

Rewrite $- (3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}})$ as $- 1 (3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}})$ .

$- \frac{- 1 (3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}})}{15} + C$

Step 17.11

Move the negative in front of the fraction.

$- - \frac{3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}}}{15} + C$

Step 17.12

Multiply $- 1$ by $- 1$ .

$1 \frac{3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}}}{15} + C$

Step 17.13

Multiply $\frac{3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}}}{15}$ by $1$ .

$\frac{3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}}}{15} + C$

Step 18

Reorder terms.

$\frac{1}{15} (3 {(1 - x^{2})}^{\frac{5}{2}} - 5 {(1 - x^{2})}^{\frac{3}{2}}) + C$