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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

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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

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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

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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

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