Calculus Examples

$\int x \cos (2 x) d x$

Step 1

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x$ and

$d v = \cos (2 x)$ .

$x (\frac{1}{2} \sin (2 x)) - \int \frac{1}{2} \sin (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine

$\frac{1}{2}$ and

$\sin (2 x)$ .

$x \frac{\sin (2 x)}{2} - \int \frac{1}{2} \sin (2 x) d x$

Step 2.2

Combine

$x$ and

$\frac{\sin (2 x)}{2}$ .

$\frac{x \sin (2 x)}{2} - \int \frac{1}{2} \sin (2 x) d x$

Step 3

Since

$\frac{1}{2}$ is constant with respect to

$x$ , move

$\frac{1}{2}$ out of the integral.

$\frac{x \sin (2 x)}{2} - (\frac{1}{2} \int \sin (2 x) d x)$

Step 4

Let

$u = 2 x$ . Then

$d u = 2 d x$ , so

$\frac{1}{2} d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 4.1

Let

$u = 2 x$ . Find

$\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate

$2 x$ .

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

Step 4.1.2

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

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

Step 4.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 \cdot 1$

Step 4.1.4

Multiply

$2$ by

$1$ .

$2$

Step 4.2

Rewrite the problem using

$u$ and

$d u$ .

$\frac{x \sin (2 x)}{2} - \frac{1}{2} \int \sin (u) \frac{1}{2} d u$

Step 5

Combine

$\sin (u)$ and

$\frac{1}{2}$ .

$\frac{x \sin (2 x)}{2} - \frac{1}{2} \int \frac{\sin (u)}{2} d u$

Step 6

Since

$\frac{1}{2}$ is constant with respect to

$u$ , move

$\frac{1}{2}$ out of the integral.

$\frac{x \sin (2 x)}{2} - \frac{1}{2} (\frac{1}{2} \int \sin (u) d u)$

Step 7

Simplify.

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Step 7.1

Multiply

$\frac{1}{2}$ by

$\frac{1}{2}$ .

$\frac{x \sin (2 x)}{2} - \frac{1}{2 \cdot 2} \int \sin (u) d u$

Step 7.2

Multiply

$2$ by

$2$ .

$\frac{x \sin (2 x)}{2} - \frac{1}{4} \int \sin (u) d u$

Step 8

The integral of

$\sin (u)$ with respect to

$u$ is

$- \cos (u)$ .

$\frac{x \sin (2 x)}{2} - \frac{1}{4} (- \cos (u) + C)$

Step 9

Rewrite

$\frac{x \sin (2 x)}{2} - \frac{1}{4} (- \cos (u) + C)$ as

$\frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4} + C$ .

$\frac{x \sin (2 x)}{2} + \frac{\cos (u)}{4} + C$

Step 10

Replace all occurrences of

$u$ with

$2 x$ .

$\frac{x \sin (2 x)}{2} + \frac{\cos (2 x)}{4} + C$

Step 11

Reorder terms.

$\frac{1}{2} x \sin (2 x) + \frac{1}{4} \cos (2 x) + C$