Multiple Choice
1.(5pts) The solution to the initial value problem
x
dy
=
dx
y
(a) y = x + C
(b) y = C +
√
(c) y = x2 + C
x
2.(5pts)
is
(d) y =
√
x2 + C
(e) y = C
π/2
sin7 t cos t dt
0
(a)
1
8
(b)
1
7
(c)
3.(5pts) The improper integral
∞
1
6
(a) converges to
3
2
(d) converges to
5
2
(e) diverges
3/2
5x + 11
is
x + 4x + 3
3
2
(b)
+
x+1 x+3
5x + 11
(e) 2
x + 4x + 3
4.(5pts) The partial fraction expansion of
2
3
+
x−1 x−3
3
4
−
(d)
x+1 x+3
(a)
(e) 2
1
dx
x
(b) converges to 2
1
(d) 1
(c) converges to
2
5
2
(c)
5
4
+
x+1 x+3
an−1
for n > 1. Then a5 =
n
1
1
(c)
(d)
120
24
5.(5pts) A sequence is defined by a1 = 1, an =
(a)
1
80
(b)
1
32
(sinh x + cosh x)3 =
e3x
1 3x
3
(b)
(c)
e
(a) (sinh x)
8
64
3
dx
√
7.(5pts) The improper integral
9 − x2
0
1
(a) converges to arcsin 3
(b) converges to 1
π
(d) converges to
(e) diverges
2
(e) 1
6.(5pts)
2
(d)
1 3x
e
2
(e) e3x
(c) converges to arcsin 3
2
8.(5pts)
x ln x dx =
1
(a) ln 8
5
4
(b) 4 ln 2 +
(c) 4 ln 2 +
1 2x
e (2x + 1) + C
2
1
(e) e2x (2x − 1) + C
4
(a) e2x (2x + 1) + C
3
4
(e) 2 ln 4 + 1
10.(5pts)
1
2
(c) e2x (x + 1) + C
(b)
1 2x
e (x − 2) + C
2
π
(a)
4
(d) 2 ln 2 −
xe2x dx =
9.(5pts)
(d)
3
4
dx
=
x − 2x + 2
π
(b)
3
2
√
11.(10pts) Evaluate
π
1
(c)
(d) arctan
2
2
Partial Credit
dx
1 + x2
3 .
12.(10pts) Solve the initial value problem
√
dy
+ y = x
dx
y(1) = 1
x
∞
13.(10pts) Evaluate
1
,
x>0
dx
.
x + 3x + 2
2
14.(10pts) Use integration by parts to evaluate
arctan x dx.
15.(10pts) Let an = 1 − ( 23 )n .
a) Show that an ≤ an+1 .
b) Show that the sequence an has an upper bound.
c) Explain why lim an exists.
n→∞
d) What number is lim an ?
n→∞
3
(e) arctan 2
Name:
Instructor-section:
Math126, Test II
March 18, 1999
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The Honor Code is in effect for this examination. All work is to be your own.
No calculators.
The exam lasts for two hours.
Be sure that your name is on every page in case pages become detached.
Be sure that you have all 13 pages of the test.
Good Luck!
Please mark your answers with an X.
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