Calculus Examples

$\int x \sqrt[3]{81 + x^{2}} d x$

Step 1

Let $u = 81 + x^{2}$ . 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 $u = 81 + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $81 + x^{2}$ .

$\frac{d}{d x} [81 + x^{2}]$

Step 1.1.2

By the Sum Rule, the derivative of $81 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [81] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [81] + \frac{d}{d x} [x^{2}]$

Step 1.1.3

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 1.1.4

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 1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 1.2

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

$\int \sqrt[3]{u} \frac{1}{2} d u$

Step 2

Combine $\sqrt[3]{u}$ and $\frac{1}{2}$ .

$\int \frac{\sqrt[3]{u}}{2} d u$

Step 3

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

$\frac{1}{2} \int \sqrt[3]{u} d u$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u}$ as $u^{\frac{1}{3}}$ .

$\frac{1}{2} \int u^{\frac{1}{3}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{1}{3}}$ with respect to $u$ is $\frac{3}{4} u^{\frac{4}{3}}$ .

$\frac{1}{2} (\frac{3}{4} u^{\frac{4}{3}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{2} (\frac{3}{4} u^{\frac{4}{3}} + C)$ as $\frac{1}{2} \cdot \frac{3}{4} u^{\frac{4}{3}} + C$ .

$\frac{1}{2} \cdot \frac{3}{4} u^{\frac{4}{3}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{4}$ .

$\frac{3}{2 \cdot 4} u^{\frac{4}{3}} + C$

Step 6.2.2

Multiply $2$ by $4$ .

$\frac{3}{8} u^{\frac{4}{3}} + C$

Step 7

Replace all occurrences of $u$ with $81 + x^{2}$ .

$\frac{3}{8} {(81 + x^{2})}^{\frac{4}{3}} + C$