Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $r^{2} - x^{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d r} [r^{n}]$ is $n r^{n - 1}$ where $n = 2$ .

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

Step 1.1.4

Since $- x^{2}$ is constant with respect to $r$ , the derivative of $- x^{2}$ with respect to $r$ is $0$ .

$2 r + 0$

Step 1.1.5

Add $2 r$ and $0$ .

$2 r$

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

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

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

Step 2.2

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

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

Step 3

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 4

Apply basic rules of exponents.

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Step 4.1

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 4.2

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 4.3

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

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Step 4.3.1

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 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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 6

Simplify.

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Step 6.1

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 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor of $2$ .

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Step 6.2.2.1

Cancel the common factor.

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

Step 6.2.2.2

Rewrite the expression.

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

Step 6.2.3

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

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

Step 7

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

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