AP CALCULUS - AB
Section Number:
4.5
MR. RECORD
Day: 52
Topics: Integration by Substitution
 Basic u-substitution
 Change of Variable Method
The role of integration by substitution is similar to that of the Chain Rule for differentiation. Recall that the
Chain Rule states
d
Given y  F (u) and u  g(x), then F (g(x))  F (g(x))  g (x) .
dx
From the definition of an antiderivative, it follows that
 F (g(x))  g(x)dx  F (g(x))  C or F (u)  C
It may be easier to think of this idea written this way instead:
du
 f (u) dx dx  F (u)  C
U-Substitution and Recognizing Patterns
Example 1:
a.
 2 x( x
2
The integrand in each of the following integrals fits the patterns above. Integrate each.
 1) dx
4
b.
 3x
2
x 3  1 dx
c.
 sec
2
x(tan x  3)dx
du
is not a perfect match. What do we do? The following Example will illustrate this issue.
dx
Example 2: The integrand in each of the following integrals still fits the patterns above. They just need a
“minor adjustment.” Integrate each.
7x2
dx
a.  x(x 2  1)6 dx
b.  2 x  1 dx
c. 
4x3  5
Sometimes
You can perform this same integration technique for trigonometric functions as well.
Example 3: Integrate each.
a.
 sin2x dx
Example 4: Integrate
b.
 csc
2

d
4
 sin 3x  cos3x dx
2
BEWARE: Some problems in the Textbook HW will not require a u-substitution method at all. Be alert.
Change of Variable Method
Here is a slightly different type of Integration by Substitution problem:
x
2 x  1 dx .
Although we will still utilize a “substitution,” we will go about it a bit differently. This method is called the
“Change of Variable” method. Your textbook (oddly enough) calls it the “Method of Example 5” prior to
Exercises 67-74. If you examine the method closely, however, it is the same as our typical “U-substitution
Method.”
Example 5: Integrate.
a.
x
2 x  1 dx
b.

2x  1
dx
x4
AP CALCULUS - AB
Section Number:
4.5
MR. RECORD
Day: 53
Topics: Integration by Substitution
 Substitution and Definite Integration
 The Accumulation Function – Part II
Substitution with Definite Integration
Example 6: Evaluate each of the following.
Note: There are two ways that you can approach these problems. Watch carefully.
5
1
3
x
2
dx
a.  x  x  1  dx
b. 
2x  1
0
1
Once again you can choose to either “back substitute” your expression in terms of x and keep the original
boundaries OR you can keep your answer in terms of u and change the boundaries. The choice is up to you.
The AP Exam likes to ask questions which require you to change the boundaries.
x
3
Example 7: Which of the following is an equivalent integral expression for
2
3
(A)
1
3 2
 u  du
  u  du
25
(B)
6
(C)
25
 u  du
25

1
3 6
1
(D)
3 6

u  2 du
(E) None of the above
2

x3  2 dx ?
The Accumulation Function- II
x
Example 7: Let F (x)   f (t )dt where f is the function graphed below (consisting of lines and a semi-circle).
0
Find the following:
a. F (0)
b. F (2)
c. F (4)
d. F (6)
e. F (1)
f. F (2)
g. F (3)
h. F (4)
i. F (4)
j. F (2)
k. F (6)
l. F ( 3)
m. On what subintervals  4,6 is F increasing and decreasing? Justify your answer.
n. Where in the interval  4,6 does F achieve its minimum value? What is the minimum value?
The information at the beginning of the problem has been recopied for your convenience.
x
Let F (x)   f (t )dt where f is the function graphed below (consisting of lines and a semi-circle).
0
o. Where in the interval  4,6 does F achieve its maximum value? What is the maximum value?
p. Where on the interval  4,6 is F concave up? Where is F concave down? Justify.
q. Where does F have points of inflection?
r. Sketch the function F.