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ERJAEE, G. (20)

Lecture 6

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

6This

**preview**shows page 1. to view the full**4 pages of the document.** MATH 2B - Lecture 6 - Substitution Rule

We begin by considering a single problem.

Ex: in(2x) dx

∫

s

We introduce a new variable u=2x, then the integral becomes:

in(2x) dx

∫

su=2x

in u dx

∫

s

Since dx is not consistent with our new variable u, we then need to translate dx to du.

u=2x → du = 2dx → dx = ½ du

in(2x) dx

∫

su=2x inu dx

∫

sdx=½ du inu du

∫

s2

1

After introduction the new variable u, and some translations, we end with the integral

inu du (− osu)

2

1∫

s=2

1c+c

cosu= − 2

1+c

cos(2x)= − 2

1+cu=2x

● This example illustrates the basic idea of u-substitution.

U-Substitution

● If u=g(x) is a differentiable function whose range is an interval I and f is

continuous on I, then:

(g(x))g(x)dx

∫

f′u=g(x)

(u) du

∫

f

u = g(x) → du = g’(x)dx

That means, if we recognize there is a composite function (“f(g(x))”), it is possible to use

the u-sub to simplify the function, such that we can solve the integral much easier.

To use u-sub, the first step is to recognize the composite function “f(g(x))”

Example:

os(x)2x dx

∫

c2+ 5

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