Calculus Examples

$x {(x^{2} + 9)}^{\frac{1}{2}}$

Step 1

Let $u = x^{2} + 9$ . 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 $u = x^{2} + 9$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 9$ .

$\frac{d}{d x} [x^{2} + 9]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 1.1.3

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

$2 x + \frac{d}{d x} [9]$

Step 1.1.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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 5

Simplify.

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Step 5.1

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 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 5.2.2

Multiply $2$ by $3$ .

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

Step 5.2.3

Cancel the common factor of $2$ and $6$ .

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Step 5.2.3.1

Factor $2$ out of $2$ .

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

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $2$ out of $6$ .

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

Step 5.2.3.2.2

Cancel the common factor.

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

Step 5.2.3.2.3

Rewrite the expression.

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

Step 6

Replace all occurrences of $u$ with $x^{2} + 9$ .

$\frac{1}{3} {(x^{2} + 9)}^{\frac{3}{2}} + C$