MTH 201 Problems Exam I, Fall 2012
Lebanese American University · Calculus III · 2 pages
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Sample for review....
Calculus III ......Hamdan, Fall 2011
1. Integrate the following
(a)
Z
xdx
4 + 9x4
(b)
R
cos¡1 xdx
2. Simplify
(a) cos
µ
tan¡1
µ 5
px
¶¶
.
(b) 10 cosh(lnx)
Evaluate the following (improper) integrals:
(a)
Z 1
0
2dx
4 + x2
(b)
Z 1
8
dx
(x ¡ 7)(x ¡ 6)
3. Determine whether the following improper integrals converge or diverge:
(a)
Z 1
1
px + 2
x5
dx
(b)
Z 3
0
dx
p3 ¡ x
(c)
Z 1
0
px
1 + x2 dx
(d)
Z 1
1
|ln x|
x
dx
(e)
Z 1
1
ln x
(x +
p
x ¡ 1)
dx
(f)
Z 1
1
ex
p
1 + x2 dx
(g)
Z 1
0
|sin x|
1 + x2 dx
4. Find the value of p > 0 for which the integral
Z 1
2
1
x(ln x)p dx converges.
5. Determine whether the improper integrals converge or diverge:
(a)
Z ¡5
¡1
8
x3 dx
(b)
Z 1
1
ex
p
1 + x2 dx
6. Determine whether the following series converges or diverges. Name the test you are using:
(a)
Z 1
1
2x2 ¡ 3x + ln x
2x ¡ 1
dx
(b)
Z 1
1
e¡x
x3 dx
(c)
Z 1
1
ln x
x1.5 dx
(d)
Z 0 e¡x
3 dx
(d)
Z º
0
dx
x + sin x
9. Sort from slowest to fastest the following functions: No need to justify:
xe,
ex,
(ln x)x ,
ln(xx),
ln(x2)
Consider the improper integral
Z 1
1
dx
(x2 + 1)3 .
(a) Determine whether it converges or diverges, using the comparison test.
(b) What can you say about
Z 1
0
dx
(x2 + 1)3 ?