4-5 INTEGRATION
BY SUBSTITUTION
MS. BATTAGLIA – AP CALCULUS
THM 4.13 ANTIDIFFERENTIATION
OF A COMPOSITE FUNCTION
Let g be a function whose range is an interval I, and let f be
a function that is continuous on I. If g is differentiable on its
domain and F is an antiderivative of f on I, then
ò f (g(x))g'(x)dx = F(g(x)) + C
Letting u = g(x) gives du = g’(x)dx and
ò
f (u)du = F(u) + C
EXPLORATION
The integrand in each of the following integrals fits the pattern
f(g(x))g’(x). Identify the pattern and use the result to evaluate the
integral.
ò 2x(x
2
+1) dx
4
ò 3x
2
x +1dx
3
The next three integrals are similar to the first three. Show how you can
multiply and divide by a constant to evaluate these integrals.
x
x
+1
dx
(
)
ò
2
4
òx
2
x +1dx
3
RECOGNIZING THE F(G(X))G’(X)
PATTERN
Find
5
cos
5x
dx
ò
MULTIPLYING AND DIVIDING BY
A CONSTANT
Find
x
x
+1
dx
(
)
ò
2
2
CHANGE IN VARIABLES
If u=g(x) then du=g’(x)dx and the integral takes the form
ò
f (g(x))g'(x)dx = ò f (u)du = F(u) + C
CHANGE OF VARIABLES
ò
2x -1 dx
CHANGE OF VARIABLES
Find
x
2x
-1
dx
ò
CHANGE OF VARIABLES
sin
3x
cos
3x
dx
ò
2
GUIDELINES FOR MAKING A
CHANGE OF VARIABLES
1. Choose a substitution u=g(x). Usually, it is best
to choose the inner part of a composite function,
such as a quantity raised to a power.
2. Compute du=g’(x)dx
3. Rewrite the integral in terms of the variable u.
4. Finding the resulting integral in terms of u.
5. Replace u by g(x) to obtain an antiderivative in
terms of x.
6. Check your answer by differentiating.
THE GENERAL POWER RULE
FOR INTEGRATION
If g is a differentiable function of x, then
ò
g(x)]
[
[ g(x)] g'(x)dx =
n+1
n
n +1
+ c, n ¹ -1
Equivalently, if u=g(x), then
n+1
u
u
du
=
+
c,
n
¹
-1
ò
n +1
n
SUBSTITUTION AND THE
GENERAL POWER RULE
3(3x
-1)
dx
ò
4
(2x
+1)(x
+
x)dx
ò
2
-4x
dx
2
ò (1- 2x )
SUBSTITUTION AND THE
GENERAL POWER RULE
3x
ò
x - 2 dx
2
3
cos
xsin
x
dx
ò
2
CHANGE OF VARIABLES FOR
DEFINITE INTEGRALS
If the function u=g(x) has a continuous derivative
on the closed interval [a,b] and f is continuous on
the range of g, then
ò
b
a
f (g(x))g'(x)dx = ò
g(b)
g(a)
f (u)du
CHANGE OF VARIABLES
ò
1
0
x ( x +1) dx
2
3
CHANGE OF VARIABLES
ò
5
1
x
dx
2x - 1
INTEGRATION OF EVEN AND
ODD FUNCTIONS
Let f be integrable on the closed interval [-a,a]
1. If f is an even function, then
2. If f is an odd function, then
ò
a
-a
ò
a
-a
a
f (x)dx = 2 ò f (x)dx
0
f (x)dx = 0
INTEGRATION OF AN ODD
FUNCTION
sin
(
ò
p /2
- p /2
3
x cos x + sin x cos x )dx
HOMEWORK
Read 4.5 Page 306
#2, 3, 7-10, 13, 29, 31, 33,
37, 39, 47, 53, 107