Calculus Examples

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

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 $d u$ .

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 2.3

Move $2$ to the left of $\sqrt{u}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Apply basic rules of exponents.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $- \frac{1}{2} (2 u^{\frac{1}{2}} + C)$ as $- \frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Multiply $2$ by $- 1$ .

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

Step 7.2.2

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

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

Step 7.2.3

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

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Step 7.2.3.1

Factor $2$ out of $- 2$ .

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

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $2$ out of $2$ .

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

Step 7.2.3.2.2

Cancel the common factor.

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

Step 7.2.3.2.3

Rewrite the expression.

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

Step 7.2.3.2.4

Divide $- 1$ by $1$ .

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

Step 8

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

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