Calculus Examples

$\int (x^{3} + x) \sqrt{x^{4} + 2 x^{2}} d x$

Step 1

Let $u = x^{4} + 2 x^{2}$ . Then $d u = (4 x^{3} + 4 x) d x$ , so $\frac{1}{4} d u = (x^{3} + x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{4} + 2 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{4} + 2 x^{2}$ .

$\frac{d}{d x} [x^{4} + 2 x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{4} + 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [2 x^{2}]$

Step 1.1.3

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

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Step 1.1.3.1

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

$4 x^{3} + 2 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$4 x^{3} + 2 (2 x)$

Step 1.1.3.3

Multiply $2$ by $2$ .

$4 x^{3} + 4 x$

Step 1.2

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

$\int \sqrt{u} \frac{1}{4} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{4}$ .

$\int \frac{\sqrt{u}}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \sqrt{u} d u$

Step 4

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

$\frac{1}{4} \int u^{\frac{1}{2}} d u$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Multiply $4$ by $3$ .

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

Step 6.2.3

Cancel the common factor of $2$ and $12$ .

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Step 6.2.3.1

Factor $2$ out of $2$ .

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

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $2$ out of $12$ .

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

Step 6.2.3.2.2

Cancel the common factor.

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

Step 6.2.3.2.3

Rewrite the expression.

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

Step 7

Replace all occurrences of $u$ with $x^{4} + 2 x^{2}$ .

$\frac{1}{6} {(x^{4} + 2 x^{2})}^{\frac{3}{2}} + C$