NAMES:
MATH 152
March 4, 2015
QUIZ 5
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
1. (3 points) Find the partial fraction decomposition (including coefficients) of
x−8
2
x − 7x + 10
x−8
A
B
x−8
+ x−2
. Then
Solution: Let x2 −7x+10 = (x−5)(x−2) = x−5
x − 8 = A(x − 2) + B(x − 5).
When x = 2, −3B = −6 so B = 2. When x = 5, 3A = −3 so A = −1. Thus
A = −1, B = 2
2. (3 points) Evaluate the integral. If the integral does not converge, write ”diverges”.
Z ∞
e−2x dx
0
Solution: By definition,
Z
∞
e
0
−2x
Z
t
e−2x dx
0
t
1 −2x = lim − e t→∞
2
0
1 −2t 1 0
= lim − e + e
t→∞
2
2
1 −2t 1
= lim − e +
t→∞
2
2
1
=
.
2
dx = lim
t→∞
NAMES:
MATH 152
March 4, 2015
3. (3 points) Evaluate the integral
x2
dx.
x+3
Solution: Since the degree of the numerator is higher than the degree of the denominator,
x2
9
use long division to find that x+3
= x − 3 + x+3
so
Z
Z
x2
9
x2
dx = x − 3 +
dx =
− 3x + 9 ln |x + 3| + C
x+3
x+3
2
Z