Calculus Examples

$\int x \sin (3 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (3 x)$ .

$x (- \frac{1}{3} \cos (3 x)) - \int - \frac{1}{3} \cos (3 x) d x$

Step 2

Simplify.

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Step 2.1

Combine $\cos (3 x)$ and $\frac{1}{3}$ .

$x (- \frac{\cos (3 x)}{3}) - \int - \frac{1}{3} \cos (3 x) d x$

Step 2.2

Combine $x$ and $\frac{\cos (3 x)}{3}$ .

$- \frac{x \cos (3 x)}{3} - \int - \frac{1}{3} \cos (3 x) d x$

Step 3

Since $- \frac{1}{3}$ is constant with respect to $x$ , move $- \frac{1}{3}$ out of the integral.

$- \frac{x \cos (3 x)}{3} - (- \frac{1}{3} \int \cos (3 x) d x)$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$- \frac{x \cos (3 x)}{3} + 1 (\frac{1}{3} \int \cos (3 x) d x)$

Step 4.2

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

$- \frac{x \cos (3 x)}{3} + \frac{1}{3} \int \cos (3 x) d x$

Step 5

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.1.2

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

$3 \frac{d}{d x} [x]$

Step 5.1.3

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

$3 \cdot 1$

Step 5.1.4

Multiply $3$ by $1$ .

$3$

Step 5.2

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

$- \frac{x \cos (3 x)}{3} + \frac{1}{3} \int \cos (u) \frac{1}{3} d u$

Step 6

Combine $\cos (u)$ and $\frac{1}{3}$ .

$- \frac{x \cos (3 x)}{3} + \frac{1}{3} \int \frac{\cos (u)}{3} d u$

Step 7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$- \frac{x \cos (3 x)}{3} + \frac{1}{3} (\frac{1}{3} \int \cos (u) d u)$

Step 8

Simplify.

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Step 8.1

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

$- \frac{x \cos (3 x)}{3} + \frac{1}{3 \cdot 3} \int \cos (u) d u$

Step 8.2

Multiply $3$ by $3$ .

$- \frac{x \cos (3 x)}{3} + \frac{1}{9} \int \cos (u) d u$

Step 9

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

$- \frac{x \cos (3 x)}{3} + \frac{1}{9} (\sin (u) + C)$

Step 10

Simplify.

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Step 10.1

Rewrite $- \frac{x \cos (3 x)}{3} + \frac{1}{9} (\sin (u) + C)$ as $- \frac{1}{3} x \cos (3 x) + \frac{1}{9} \sin (u) + C$ .

$- \frac{1}{3} x \cos (3 x) + \frac{1}{9} \sin (u) + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $x$ and $\frac{1}{3}$ .

$- \frac{x}{3} \cos (3 x) + \frac{1}{9} \sin (u) + C$

Step 10.2.2

Combine $\cos (3 x)$ and $\frac{x}{3}$ .

$- \frac{\cos (3 x) x}{3} + \frac{1}{9} \sin (u) + C$

Step 11

Replace all occurrences of $u$ with $3 x$ .

$- \frac{\cos (3 x) x}{3} + \frac{1}{9} \sin (3 x) + C$

Step 12

Reorder factors in $\frac{\cos (3 x) x}{3}$ .

$- \frac{1}{3} x \cos (3 x) + \frac{1}{9} \sin (3 x) + C$

Step 13

Simplify.

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Step 13.1

Combine $x$ and $\frac{1}{3}$ .

$- \frac{x}{3} \cos (3 x) + \frac{1}{9} \sin (3 x) + C$

Step 13.2

Combine $\cos (3 x)$ and $\frac{x}{3}$ .

$- \frac{\cos (3 x) x}{3} + \frac{1}{9} \sin (3 x) + C$

Step 14

Reorder factors in $\frac{\cos (3 x) x}{3}$ .

$- \frac{1}{3} x \cos (3 x) + \frac{1}{9} \sin (3 x) + C$