Calculus Examples

$\int 18 x^{2} \sqrt[4]{6 x^{3} + 5} d x$

Step 1

Since $18$ is constant with respect to $x$ , move $18$ out of the integral.

$18 \int x^{2} \sqrt[4]{6 x^{3} + 5} d x$

Step 2

Let $u = 6 x^{3} + 5$ . Then $d u = 18 x^{2} d x$ , so $\frac{1}{18} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 6 x^{3} + 5$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $6 x^{3} + 5$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $6 x^{3} + 5$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [5]$ .

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

Step 2.1.3

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 2.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

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

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 $n = 3$ .

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

Step 2.1.3.3

Multiply $3$ by $6$ .

$18 x^{2} + \frac{d}{d x} [5]$

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

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

$18 x^{2} + 0$

Step 2.1.4.2

Add $18 x^{2}$ and $0$ .

$18 x^{2}$

Step 2.2

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

$18 \int \sqrt[4]{u} \frac{1}{18} d u$

Step 3

Combine $\sqrt[4]{u}$ and $\frac{1}{18}$ .

$18 \int \frac{\sqrt[4]{u}}{18} d u$

Step 4

Since $\frac{1}{18}$ is constant with respect to $u$ , move $\frac{1}{18}$ out of the integral.

$18 (\frac{1}{18} \int \sqrt[4]{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{18}$ and $18$ .

$\frac{18}{18} \int \sqrt[4]{u} d u$

Step 5.1.2

Cancel the common factor of $18$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{18}{18} \int \sqrt[4]{u} d u$

Step 5.1.2.2

Rewrite the expression.

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

Step 5.1.3

Multiply $\int \sqrt[4]{u} d u$ by $1$ .

$\int \sqrt[4]{u} d u$

Step 5.2

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

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

Step 6

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

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

Step 7

Replace all occurrences of $u$ with $6 x^{3} + 5$ .

$\frac{4}{5} {(6 x^{3} + 5)}^{\frac{5}{4}} + C$