Calculus Examples

$\int x^{5} \sqrt{2 x^{6} + 3} d x$

Step 1

Let $u = 2 x^{6} + 3$ . Then $d u = 12 x^{5} d x$ , so $\frac{1}{12} d u = x^{5} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x^{6} + 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{6} + 3$ .

$\frac{d}{d x} [2 x^{6} + 3]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{6} + 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{6}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x^{6}] + \frac{d}{d x} [3]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{6}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{6}$ with respect to $x$ is $2 \frac{d}{d x} [x^{6}]$ .

$2 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [3]$

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 $n = 6$ .

$2 (6 x^{5}) + \frac{d}{d x} [3]$

Step 1.1.3.3

Multiply $6$ by $2$ .

$12 x^{5} + \frac{d}{d x} [3]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$12 x^{5} + 0$

Step 1.1.4.2

Add $12 x^{5}$ and $0$ .

$12 x^{5}$

Step 1.2

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

$\int \sqrt{u} \frac{1}{12} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{12}$ .

$\int \frac{\sqrt{u}}{12} d u$

Step 3

Since $\frac{1}{12}$ is constant with respect to $u$ , move $\frac{1}{12}$ out of the integral.

$\frac{1}{12} \int \sqrt{u} d u$

Step 4

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

$\frac{1}{12} \int u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{12} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{12} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $\frac{1}{12} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{12} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 6.2

Rewrite $\frac{1}{12} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ as $\frac{1}{18} u^{\frac{3}{2}} + C$ .

$\frac{1}{18} u^{\frac{3}{2}} + C$

Step 7

Replace all occurrences of $u$ with $2 x^{6} + 3$ .

$\frac{1}{18} {(2 x^{6} + 3)}^{\frac{3}{2}} + C$