Definite Integrals 249
11.6 Practice Problems
Part A The use of a calculator is not
allowed.
Evaluate the following definite integrals.
1.
∫ 0
− 1
(1+x−x^3 )dx
2.
∫ 11
6
(x− 2 )^1 /^2 dx
3.
∫ 3
1
t
t+ 1
dt
4.
∫ 6
0
∣∣
x− 3
∣∣
dx
- If
∫k
0
(6x−1)dx=4, findk.
6.
∫π
0
sinx
√
1 +cosx
dx
- Iff′(x)=g(x) andgis a continuous
function for all real values of∫ x, express
2
1
g(4x)dxin terms off.
8.
∫ln 3
ln 2
10 exdx
9.
∫e 2
e
1
t+ 3
dt
- Iff(x)=
∫x
−π/ 4
tan^2 (t)dt, findf′
(
π
6
)
.
11.
∫ 1
− 1
4 xex^2 dx
12.
∫π
−π
(
cosx−x^2
)
dx
Part B Calculators are allowed.
- Findkif
∫ 2
0
(
x^3 +k
)
dx=10.
- Evaluate
∫ 3. 1
− 1. 2
2 θcosθdθto the nearest
100th.
- Ify=
∫x 3
1
√
t^2 + 1 dt, find
dy
dx
.
- Use a midpoint Riemann sum with four
subdivisions of equal length to find the
approximate value of
∫ 8
0
(
x^3 + 1
)
dx.
- Given
∫ 2
− 2
g(x)dx= 8
and
∫ 2
0
g(x)dx=3, find
(a)
∫ 0
− 2
g(x)dx
(b)
∫− 2
2
g(x)dx
(c)
∫− 2
0
5 g(x)dx
(d)
∫ 2
− 2
2 g(x)dx
- Evaluate
∫ 1 / 2
0
√dx
1 −x^2
.
- Find
dy
dx
ify=
∫sinx
cosx
(2t+1)dt.
- Let fbe a continuous function defined on
[0, 30] with selected values as shown below:
x 0 5 10 15 20 25 30
f(x) 1.4 2.6 3.4 4.1 4.7 5.2 5.7