Calculus Examples

$\int \sqrt{1 + 4 x^{3}} (12 x^{2}) d x$

Step 1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int \sqrt{1 + 4 x^{3}} (x^{2}) d x$

Step 2

Let $u = 1 + 4 x^{3}$ . 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 $u = 1 + 4 x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $1 + 4 x^{3}$ .

$\frac{d}{d x} [1 + 4 x^{3}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $1 + 4 x^{3}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [4 x^{3}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [4 x^{3}]$

Step 2.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ 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 $d u$ .

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

Step 3

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

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

Step 4

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

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

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 $12$ .

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

Step 5.1.2

Cancel the common factor of $12$ .

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Step 5.1.2.1

Cancel the common factor.

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

Step 5.1.2.2

Rewrite the expression.

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

Step 5.1.3

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

$\int \sqrt{u} d u$

Step 5.2

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

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

Step 6

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

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

Step 7

Replace all occurrences of $u$ with $1 + 4 x^{3}$ .

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