Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $3 x$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $3$ by $1$ .

$3$

Step 4.2

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Multiply $3$ by $3$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Reorder terms.

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

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