Calculus Examples

$\int 2 x^{2} \sqrt[4]{5 + 4 x^{3}} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x^{2} \sqrt[4]{5 + 4 x^{3}} d x$

Step 2

Let $u = 5 + 4 x^{3}$ . Then $d u = 12 x^{2} d x$ , so $\frac{1}{12} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 5 + 4 x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $5 + 4 x^{3}$ .

$\frac{d}{d x} [5 + 4 x^{3}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $5 + 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [4 x^{3}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [4 x^{3}]$

Step 2.1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 x^{3}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$0 + 4 \frac{d}{d x} [x^{3}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 + 4 (3 x^{2})$

Step 2.1.3.3

Multiply $3$ by $4$ .

$0 + 12 x^{2}$

Step 2.1.4

Add $0$ and $12 x^{2}$ .

$12 x^{2}$

Step 2.2

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

$2 \int \sqrt[4]{u} \frac{1}{12} d u$

Step 3

Combine $\sqrt[4]{u}$ and $\frac{1}{12}$ .

$2 \int \frac{\sqrt[4]{u}}{12} d u$

Step 4

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

$2 (\frac{1}{12} \int \sqrt[4]{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{12}$ and $2$ .

$\frac{2}{12} \int \sqrt[4]{u} d u$

Step 5.1.2

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

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Step 5.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{12} \int \sqrt[4]{u} d u$

Step 5.1.2.2

Cancel the common factors.

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Step 5.1.2.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 1}{2 \cdot 6} \int \sqrt[4]{u} d u$

Step 5.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 6} \int \sqrt[4]{u} d u$

Step 5.1.2.2.3

Rewrite the expression.

$\frac{1}{6} \int \sqrt[4]{u} d u$

Step 5.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{u}$ as $u^{\frac{1}{4}}$ .

$\frac{1}{6} \int u^{\frac{1}{4}} d u$

Step 6

By the Power Rule, the integral of $u^{\frac{1}{4}}$ with respect to $u$ is $\frac{4}{5} u^{\frac{5}{4}}$ .

$\frac{1}{6} (\frac{4}{5} u^{\frac{5}{4}} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{1}{6} (\frac{4}{5} u^{\frac{5}{4}} + C)$ as $\frac{1}{6} \cdot \frac{4}{5} u^{\frac{5}{4}} + C$ .

$\frac{1}{6} \cdot \frac{4}{5} u^{\frac{5}{4}} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{1}{6}$ by $\frac{4}{5}$ .

$\frac{4}{6 \cdot 5} u^{\frac{5}{4}} + C$

Step 7.2.2

Multiply $6$ by $5$ .

$\frac{4}{30} u^{\frac{5}{4}} + C$

Step 7.2.3

Cancel the common factor of $4$ and $30$ .

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Step 7.2.3.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{30} u^{\frac{5}{4}} + C$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $2$ out of $30$ .

$\frac{2 \cdot 2}{2 \cdot 15} u^{\frac{5}{4}} + C$

Step 7.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 15} u^{\frac{5}{4}} + C$

Step 7.2.3.2.3

Rewrite the expression.

$\frac{2}{15} u^{\frac{5}{4}} + C$

Step 8

Replace all occurrences of $u$ with $5 + 4 x^{3}$ .

$\frac{2}{15} {(5 + 4 x^{3})}^{\frac{5}{4}} + C$