248 STEP 4. Review the Knowledge You Need to Score High
11.5 Rapid Review
- Evaluate
∫x
π/ 2
costdt.
Answer: sint]xx/ 2 =sinx−sin
(
π
2
)
=sinx−1.
- Evaluate
∫ 1
0
1
x+ 1
dx.
Answer: ln(x+1)]^10 =ln 2−ln 1=ln 2.
- IfG(x)=
∫x
0
(2t+1)^3 /^2 dt, findG′(4).
Answer: G′(x)=(2x+1)^3 /^2 andG′(4)= 93 /^2 =27.
- If
∫k
1
2 xdx=8, findk.
Answer: x^2
]k
1 =^8 ⇒k
(^2) − 1 = 8 ⇒k=±3.
- IfG(x) is an antiderivative of (ex+1) andG(0)=0, findG(1).
Answer: G(x)=ex+x+C
G(0)=e^0 + 0 +C= 0 ⇒C=−1.
G(1)=e^1 + 1 − 1 =e. - IfG′(x)=g(x), express
∫ 2
0
g(4x)dxin terms ofG(x).
Answer: Letu= 4 x;
du
4
=dx.
∫
g(u)
du
4
=
1
4
G(u).Thus,
∫ 2
0
g( 4 x)dx=
1
4
G( 4 x)
] 2
0
=
1
4
[G(8)−G(0)].
7.
∫∞
1
dx
x^2
Answer:
∫∞
1
dx
x^2
=limn→∞
∫n
1
dx
x^2
=limn→∞
[
− 1
x
]n
1
=limn→∞
[
− 1
n
+ 1
]
=1.
8.
∫ 1
0
dx
√
x
Answer:
∫ 1
0
√dx
x
=limk→ 0 +
∫ 1
k
√dx
x
=limk→ 0 +
[
2
√
x
] 1
k=klim→ 0 +
[
2 − 2
√
k
]
=2.