u-Substitution Examples
Example 1. Evaluate
Z
t5
p
t2 − 9 dt.
We let u = t2 − 9 (which implies that t2 = u + 9), then
du = 2t dt
1
du = t dt
2
Then:
Z
t5
p
t2 − 9 dt =
Z
Z
=
t4
p
t2 − 9t dt
2
(t2 )
p
t2 − 9t dt
√ 1
(u + 9)2 u du
2
Z
1
u2 + 18u + 81 u1/2 du
2
Z
1 5/2
u + 18u3/2 + 81u1/2 du
2
1 2 7/2 36 5/2 162 3/2
u + u +
u
+C
2 7
5
3
1 7/2 18 5/2 81 3/2
u + u + u +C
7
5
3
1 2
18 2
7/2
(t − 9)
+
(t − 9)5/2 + 27(t2 − 9)3/2 + C
7
5
Z
=
=
=
=
=
=
Example 2. Evaluate
Z
x2 − 1
√
dx.
2x − 1
We let u = 2x − 1 (which implies that x = 21 (u + 1)), then
du = 2 dx
1
du = dx
2
1
Then:
Z
x2 − 1
√
dx =
2x − 1
Z
(x2 − 1)(2x − 1)−1/2 dx
!
2
Z 1
1
=
(u + 1) − 1 u−1/2 du
2
2
Z 1
1
=
(u + 1)2 − 1 u−1/2 du
4
2
Z
1
1
=
(u + 1)2 − 4 u−1/2 du
4
2
Z
1
2
−1/2
=
(u + 2u + 1 − 4)u
du
8
Z
1
(u2 + 2u − 3)u−1/2 du
=
8
Z
1
u3/2 + 2u1/2 − 3u−1/2 du
=
8
1 2 5/2 4 3/2
=
u + u − 6u1/2 + C
8 5
3
1
1
3
=
(2x − 1)5/2 + (2x − 1)3/2 − (2x − 1)1/2 + C
20
6
4
Example 3. Evaluate
Z9
√
We let u = 1 +
√
1
1
x (1 +
√
2
dx.
x)
x, then
1
du = √ dx
2 x
1
2 du = √ dx
x
√
√
Also, when x = 1, u = 1 + 1 = 2, and when x = 9, u = 1 + 9 = 4. Then:
Z9
√
1
√
dx =
x(1 + x)2
Z9
(1 +
1
√
x)2
1
1
Z4
=
1
2 du
u2
2
Z4
=
2u−2 du
2
4
= −2u−1 2
4
−2
=
u 2
1 1
= −2
−
4 2
1
=
2
Example 4. Evaluate
Z4
√
x
dx.
2x + 1
0
2
1
√ dx
x
We let u = 2x + 1 (which implies that x = 21 (u − 1)), then
du = 2 dx
1
du = dx
2
Also, when x = 0, u = 1, and when x = 4, u = 9.
Then:
Z4
x
√
dx =
2x + 1
Z4
x(2x + 1)−1/2 dx
0
0
Z9
=
1
1
(u − 1)(u−1/2 ) du
2
2
1
1
=
4
Z9
u1/2 − u−1/2 du
1
9
1 2 3/2
1/2
u − 2u
4 3
1
1
2 3/2
2 3/2
1/2
1/2
·9 −2·9
−
·1 −2·1
4
3
3
1
2
18 − 6 − + 2
4
3
1 40
·
4 3
10
3
=
=
=
=
=
Example 5. Evaluate
Z
sin(2u) cos(2u) du.
We let w = sin(2u), then
du = 2 cos(2u) dx
1
du = cos(2u) dx
2
Then:
Z
Z
1
dw
2
w2 1
=
+ C1
2 2
1
= sin2 (2u) + C1
4
sin(2u) cos(2u) du =
w
Alternatively, we can let w = cos(2u), then
du = −2 sin(2u) dx
−
1
du = cos(2u) dx
2
3
Then:
Z
−1
dw
2
w2 −1
=
+ C2
2 2
1
= − cos2 (2u) + C2
4
1
1 1
1
2
2
2
= − 1 − sin (2u) + C2 = − + sin (2u) + C2 = sin (2u) + C1
4
4 4
4
Z
sin(2u) cos(2u) du =
w
4