Definite Integration with u-Substitution - Classwork
When you have to find a definite integral involving u-substitution, it is often convenient to determine the limits of
integration in terms of the variable u, rather than having to integrate and switch back to x and then substitute. This
is called “changing the limits.”
2
2
! x( x + 1) dx
2
Example 1)
Start off by finding u ___________
x = 2, u = _________
0
2
2
1
du = ________ Do we have it?
x = 0, u = _________
2 x x 2 + 1 dx
2 0
Now write everything in terms of u and calculate. It is no longer necessary to switch back to x.
! (
)
1
Example 2)
5
!x
2
1 " x dx
Example 3)
0
#
Example 4)
0
sin 2 x dx
Example 5)
0
!
0
x +4
dx
! sin x
cos x dx
0
#6
Example 6)
x
2
# 2
4
!
!
2
(1 + sin x cos x ) dx
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Example 7)
!
1 " x 2 dx
"2
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Stu Schwartz
Definite Integration with u-Substitution - Homework
Find the values of the following definite integrals. Verify using your calculator. Some will use u-substitution,
others will not.
4
2
! ( x " 1) dx
3
1.
2.
"2
1
! (1 " cos 2 x ) dx
5.
5
)
3.
!
8.
2
+ 1 dx
3
6.
!
0
dt
3
!
x " x dx
9.
2t + 1
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!
2
9 " x 2 dx
x
(
)
x2 " 3
2
dx
#2
#2
11.
!x
0
0
4
! sin(2 x ) dx
0
4
x " 4 dx
0
10.
! 2 x( x
2
0
0
7.
)
x " 1 dx
0
# 12
4.
! x(
#3
! cos t sin t dt
3
0
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12.
! t sin(# " t ) dt
2
0
Stu Schwartz
#2
! t sin(# " t ) dt
13.
14.
0
!
0
f ( x) dx =
11
and
3
17.
!
"2
!
0
f ( x) dx =
!
"2
!
2
f ( x) dx
21.
!
0
dx (Be careful!)
6
0
" f ( x) dx
22.
!
"2
3 f ( x) dx
23.
! f (3x ) dx
0
6
! f ( x ) dx = 15 , f ( x) is an odd function (symmetric to the origin), find the following:
0
2
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2
"4
0
"2
f ( x) dx
! x1
! f ( x ) dx = 15 , f ( x) is an even function (symmetric to the y - axis), find the following:
0
24.
18.
6
20.
11
and
3
ax 2 + b dx
4
9 " x 2 dx
2
f ( x) dx
2
If
!
!x
0
0
0
19.
15.
1 " cos x dx
4
sin x
!2 x dx
# 4
2
If
! cos x
2
0
#2
16.
1
# 3
2
25.
!
"2
2
f ( x) dx
26.
!
0
2
0
" f ( x) dx
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27.
!
"2
3 f ( x) dx
28.
! f (3x ) dx
0
Stu Schwartz