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How to Do Integration by Substitution


Substitution in integration is the chain rule reversed. The formula is as follows:

Formula

Substitution

f(g(x)) g(x)dx =f(u)du = F(u) = F(g(x))

You use substitution when the expression contains two functions where one of the functions is the derivative of the other. Choose u as the function that is not differentiated! Next, you differentiate both sides with respect to x so that you have du dx. Then, you multiply by dx on both sides of the equation to cancel the dx in the denominator, such that you have solved for du.

Example 1

2xex2dx = eudu = eu + C = ex2 + C

2xex2dx = eudu = eu + C = ex2 + C

*

u = x2 du dx = 2x du = 2xdx

Example 2

Compute 2x + 1 x2 + xdx

2x + 1 x2 + xdx = 1 udu = ln |u| + C = ln |x2 + x| + C

*

u = x2 + x du dx = 2x + 1 du = 2x + 1dx

Example 3

Compute sin x cos 2xdx

sin x cos 2xdx = 1 u2du = u2du = u1 + C = 1 u + C = 1 cos x + C

*

u = cos x du dx = sin x du = sin xdx

Example 4

Compute cos xesin xdx

cos xesin xdx = eudu = eu + C = esin x + C

*

u = sin x du dx = cos x du = cos xdx

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