Calculus Examples

$\int x \sin (8 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (8 x)$ .

$x (- \frac{1}{8} \cos (8 x)) - \int - \frac{1}{8} \cos (8 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\cos (8 x)$ and $\frac{1}{8}$ .

$x (- \frac{\cos (8 x)}{8}) - \int - \frac{1}{8} \cos (8 x) d x$

Step 2.2

Combine $x$ and $\frac{\cos (8 x)}{8}$ .

$- \frac{x \cos (8 x)}{8} - \int - \frac{1}{8} \cos (8 x) d x$

Step 3

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

$- \frac{x \cos (8 x)}{8} - (- \frac{1}{8} \int \cos (8 x) d x)$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$- \frac{x \cos (8 x)}{8} + 1 (\frac{1}{8} \int \cos (8 x) d x)$

Step 4.2

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

$- \frac{x \cos (8 x)}{8} + \frac{1}{8} \int \cos (8 x) d x$

Step 5

Let $u = 8 x$ . 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 5.1

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

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Step 5.1.1

Differentiate $8 x$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

$8 \cdot 1$

Step 5.1.4

Multiply $8$ by $1$ .

$8$

Step 5.2

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

$- \frac{x \cos (8 x)}{8} + \frac{1}{8} \int \cos (u) \frac{1}{8} d u$

Step 6

Combine $\cos (u)$ and $\frac{1}{8}$ .

$- \frac{x \cos (8 x)}{8} + \frac{1}{8} \int \frac{\cos (u)}{8} d u$

Step 7

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

$- \frac{x \cos (8 x)}{8} + \frac{1}{8} (\frac{1}{8} \int \cos (u) d u)$

Step 8

Simplify.

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Step 8.1

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

$- \frac{x \cos (8 x)}{8} + \frac{1}{8 \cdot 8} \int \cos (u) d u$

Step 8.2

Multiply $8$ by $8$ .

$- \frac{x \cos (8 x)}{8} + \frac{1}{64} \int \cos (u) d u$

Step 9

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$- \frac{x \cos (8 x)}{8} + \frac{1}{64} (\sin (u) + C)$

Step 10

Simplify.

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Step 10.1

Rewrite $- \frac{x \cos (8 x)}{8} + \frac{1}{64} (\sin (u) + C)$ as $- \frac{1}{8} x \cos (8 x) + \frac{1}{64} \sin (u) + C$ .

$- \frac{1}{8} x \cos (8 x) + \frac{1}{64} \sin (u) + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $x$ and $\frac{1}{8}$ .

$- \frac{x}{8} \cos (8 x) + \frac{1}{64} \sin (u) + C$

Step 10.2.2

Combine $\cos (8 x)$ and $\frac{x}{8}$ .

$- \frac{\cos (8 x) x}{8} + \frac{1}{64} \sin (u) + C$

Step 11

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

$- \frac{\cos (8 x) x}{8} + \frac{1}{64} \sin (8 x) + C$

Step 12

Reorder factors in $\frac{\cos (8 x) x}{8}$ .

$- \frac{1}{8} x \cos (8 x) + \frac{1}{64} \sin (8 x) + C$