5 Steps to a 5 AP Calculus BC 2019

Integration 213

10.2 Integration by U-Substitution

Main Concepts:The U-Substitution Method, U-Substitution and Algebraic

Functions, U-Substitution and Trigonometric Functions,

U-Substitution and Inverse Trigonometric Functions, U-Substitution

and Logarithmic and Exponential Functions

The U-Substitution Method

The Chain Rule for Differentiation

d

dx

F(g(x))=f(g(x))g′(x), whereF′= f

The Integral of a Composite Function

Iff(g(x)) andf′are continuous andF′=f, then

∫

f(g(x))g′(x)dx=F(g(x))+C.

Making a U-Substitution

Letu=g(x), thendu=g′(x)dx

∫

f(g(x))g′′(x)dx=

∫

f(u)du=F(u)+C=F(g(x))+C.

Procedure for Making a U-Substitution

STRATEGY

Steps:

1. Given f(g(x)); letu=g(x).


2. Differentiate:du=g′(x)dx.


3. Rewrite the integral in terms ofu.


4. Evaluate the integral.


5. Replaceubyg(x).


6. Check your result by taking the derivative of the answer.

U-Substitution and Algebraic Functions

Another Form of the Integral of a Composite Function

Iff is a differentiable function, then

∫

(f(x))nf′(x)dx=

(f(x))n+^1

n+ 1

+C,n=−/ 1.

Making a U-Substitution

Letu=f(x); thendu= f′(x)dx.

∫

(f(x))nf′(x)dx=

∫

undu=

un+^1

n+ 1

+C=

(f(x))n+^1

n+ 1

+C,n/=− 1