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:
- Given f(g(x)); letu=g(x).
- Differentiate:du=g′(x)dx.
- Rewrite the integral in terms ofu.
- Evaluate the integral.
- Replaceubyg(x).
- 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