Calculus Examples

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

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

$d u$ .

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

Combine

$\sqrt{u}$ and

$\frac{1}{2}$ .

$\int - \frac{\sqrt{u}}{2} d u$

Step 3

Since

$- 1$ is constant with respect to

$u$ , move

$- 1$ out of the integral.

$- \int \frac{\sqrt{u}}{2} d u$

Step 4

Since

$\frac{1}{2}$ is constant with respect to

$u$ , move

$\frac{1}{2}$ out of the integral.

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

Step 5

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{u}$ as

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

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

Step 6

By the Power Rule, the integral of

$u^{\frac{1}{2}}$ with respect to

$u$ is

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

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

Step 7

Simplify.

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Step 7.1

Rewrite

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

$- \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$- \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply

$\frac{2}{3}$ by

$\frac{1}{2}$ .

$- \frac{2}{3 \cdot 2} u^{\frac{3}{2}} + C$

Step 7.2.2

Multiply

$3$ by

$2$ .

$- \frac{2}{6} u^{\frac{3}{2}} + C$

Step 7.2.3

Cancel the common factor of

$2$ and

$6$ .

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Step 7.2.3.1

Factor

$2$ out of

$2$ .

$- \frac{2 (1)}{6} u^{\frac{3}{2}} + C$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor

$2$ out of

$6$ .

$- \frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 7.2.3.2.2

Cancel the common factor.

$- \frac{2 \cdot 1}{2 \cdot 3} u^{\frac{3}{2}} + C$

Step 7.2.3.2.3

Rewrite the expression.

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

Step 8

Replace all occurrences of

$u$ with

$1 - x^{2}$ .

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