Colorful House of Math logo
Menu

Functions

Menu
How to Use the Formula for Integration by Parts

Integration by parts is simply the product rule reversed. The formula is as follows:

Formula

Integration by Parts

uvdx = uv uvdx

Note! In exercises with integration by parts, you should choose ex as v and ln(x) as u.

Example 1

3xexdx = 3xex 3exdx = 3xex 3ex + C = 3ex(x 1) + C

*

u = 3xv = ex u = 3 v = ex

Example 2

Find the function F such that F(x) = 4x3 + 1 x and F(1) = 2

F(x) = 4x3 + 1 xdx = x4 + ln |x| + C

Furthermore, given that F(1) = 2:

2 = F(1) = 14 + ln |1| + C = 1 + 0 + C, C = 1

Then, F(x) = x4 + ln |x| + 1.

Example 3

Compute the integral sin 2xdx

sin 2xdx = sin x sin xdx = sin x cos x = + cos 2xdx = sin x cos x = + 1 sin 2xdx = sin x cos x = + x sin 2xdx

sin 2xdx = sin x sin xdx = sin x cos x + cos 2xdx = sin x cos x + 1 sin 2xdx = sin x cos x + x sin 2xdx

*

u = sin xv = sin x u = cos xv = cos x

This gives you an equation that you solve for sin 2xdx:

sin 2xdx = sin x cos x = + x sin 2xdx

2 sin 2xdx 2 = sin x cos x + x 2 sin 2xdx = 1 2 sin x cos x + x 2 + C

sin 2xdx = sin x cos x + x sin 2xdx 2 sin 2xdx = sin x cos x + x| : 2 sin 2xdx = 1 2 sin x cos x + x 2 + C

Example 4

Compute cos(2x) sin(2x)dx

= cos(2x) sin(2x)dx = 1 2 cos 2(2x) sin(2x) cos(2x)dx

cos(2x) sin(2x)dx = 1 2 cos 2(2x) sin(2x) cos(2x)dx

*

u = cos(2x) v = sin(2x) u = 2 sin(2x)v = 1 2 cos(2x)

You now solve this expression as an equation with respect to cos(2x) sin(2x)dx:

cos(2x) sin(2x)dx = 1 2 cos 2(2x) sin(2x) cos(2x)dx 2cos(2x) sin(2x)dx = 1 2 cos 2(2x)| ÷2 cos(2x) sin(2x)dx = 1 4 cos 2(2x) + C

cos(2x) sin(2x)dx = 1 2 cos 2(2x) sin(2x) cos(2x)dx 2 cos(2x) sin(2x)dx = 1 2 cos 2(2x)| ÷ 2 cos(2x) sin(2x)dx = 1 4 cos 2(2x) + C

Example 5

Compute ex (x2 + 3x 4) dx

= ex (x2 + 3x 4) dx = ex (x2 + 3x 4) ex(2x + 3)dx = ex (x2 + 3x 4) (ex(2x + 3) 2exdx ) = ex (x2 + 3x 4) ex(2x + 3) + 2ex + C = ex (x2 + x 5) + C

ex (x2 + 3x 4) dx = ex (x2 + 3x 4) ex(2x + 3)dx = ex (x2 + 3x 4) (ex(2x + 3) 2exdx) = ex (x2 + 3x 4) ex(2x + 3) + 2ex + C = ex (x2 + x 5) + C

*

u = x2 + 3x 4v = ex u = 2x + 3 v = ex

**

z = 2x + 3w = ex z = 2 w = ex

Globe AI
AI
How can I help you?
Beta