Calculus Examples

$\int \sqrt{8 x + 1} d x$

Step 1

Let $u = 8 x + 1$ . Then $d u = 8 d x$ , so $\frac{1}{8} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 8 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $8 x + 1$ .

$\frac{d}{d x} [8 x + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $8 x + 1$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 1.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

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 $n = 1$ .

$8 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$8 + 0$

Step 1.1.4.2

Add $8$ and $0$ .

$8$

Step 1.2

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

$\int \sqrt{u} \frac{1}{8} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{8}$ .

$\int \frac{\sqrt{u}}{8} d u$

Step 3

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int \sqrt{u} d u$

Step 4

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

$\frac{1}{8} \int u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{8} (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{8} (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $\frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{8} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{8}$ by $\frac{2}{3}$ .

$\frac{2}{8 \cdot 3} u^{\frac{3}{2}} + C$

Step 6.2.2

Multiply $8$ by $3$ .

$\frac{2}{24} u^{\frac{3}{2}} + C$

Step 6.2.3

Cancel the common factor of $2$ and $24$ .

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Step 6.2.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{24} u^{\frac{3}{2}} + C$

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $2$ out of $24$ .

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

Step 6.2.3.2.2

Cancel the common factor.

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

Step 6.2.3.2.3

Rewrite the expression.

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

Step 7

Replace all occurrences of $u$ with $8 x + 1$ .

$\frac{1}{12} {(8 x + 1)}^{\frac{3}{2}} + C$