Calculus Examples

$f (x) = x \sqrt{7 x^{2} + 2}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int x \sqrt{7 x^{2} + 2} d x$

Step 3

Let $u = 7 x^{2} + 2$ . Then $d u = 14 x d x$ , so $\frac{1}{14} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 7 x^{2} + 2$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $7 x^{2} + 2$ .

$\frac{d}{d x} [7 x^{2} + 2]$

Step 3.1.2

By the Sum Rule, the derivative of $7 x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [2]$

Step 3.1.3

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 3.1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$

Step 3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$7 (2 x) + \frac{d}{d x} [2]$

Step 3.1.3.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [2]$

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$14 x + 0$

Step 3.1.4.2

Add $14 x$ and $0$ .

$14 x$

Step 3.2

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

$\int \sqrt{u} \frac{1}{14} d u$

Step 4

Combine $\sqrt{u}$ and $\frac{1}{14}$ .

$\int \frac{\sqrt{u}}{14} d u$

Step 5

Since $\frac{1}{14}$ is constant with respect to $u$ , move $\frac{1}{14}$ out of the integral.

$\frac{1}{14} \int \sqrt{u} d u$

Step 6

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

$\frac{1}{14} \int u^{\frac{1}{2}} d u$

Step 7

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

$\frac{1}{14} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $\frac{1}{14} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $\frac{1}{14} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{14} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{1}{14}$ by $\frac{2}{3}$ .

$\frac{2}{14 \cdot 3} u^{\frac{3}{2}} + C$

Step 8.2.2

Multiply $14$ by $3$ .

$\frac{2}{42} u^{\frac{3}{2}} + C$

Step 8.2.3

Cancel the common factor of $2$ and $42$ .

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Step 8.2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{42} u^{\frac{3}{2}} + C$

Step 8.2.3.2

Cancel the common factors.

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Step 8.2.3.2.1

Factor $2$ out of $42$ .

$\frac{2 \cdot 1}{2 \cdot 21} u^{\frac{3}{2}} + C$

Step 8.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 21} u^{\frac{3}{2}} + C$

Step 8.2.3.2.3

Rewrite the expression.

$\frac{1}{21} u^{\frac{3}{2}} + C$

Step 9

Replace all occurrences of $u$ with $7 x^{2} + 2$ .

$\frac{1}{21} {(7 x^{2} + 2)}^{\frac{3}{2}} + C$

Step 10

The answer is the antiderivative of the function $f (x) = x \sqrt{7 x^{2} + 2}$ .

$F (x) =$ $\frac{1}{21} {(7 x^{2} + 2)}^{\frac{3}{2}} + C$