MATH 145
WINTER 2008
SAMPLE EXAM II โ ANSWERS
1. (a) HINT: Substitution: let u= 1 โx.
ANSWER: 2
3
(b) HINT: Substitution: let u= 1 + ln x.
ANSWER: 7
3
(c) HINT: Substituion: let u= sin x.
ANSWER: 2โsin x+C
(d) HINT: Partial fractions: xโ2
x2(xโ1) =1
x+2
x2โ1
xโ1
ANSWER: ln |x| โ 2
xโln |xโ1|+C
(e) HINT: First evaluate the integral from 0 to band then take the limit as bapproaches
infinity.
ANSWER: 1
2
(f) HINT: First evaluate the integral from bto 1 and then take the limit as bapproaches 0
(from the positive direction).
ANSWER: 4
3
(g) HINT: First evaluate the integral from bto 1 and then take the limit as bapproaches 0
(from the positive direction).
ANSWER: This improper integral diverges.
2. (a) HINT: w(3.25) โw(0) = Z3.25
0
(8t+ 1)2/3dt. Do substitution with u= 8t+ 1.
ANSWER: 3
40 (275/3โ1) = 18.15 grams
(b) HINT: Compute w(10) โw(2) = Z10
2
(8t+ 1)2/3dt, substitute 29 for w(2), and solve for
w(10).
ANSWER: 134.30 grams
3. HINT: The area is Z4
โ1
(x+ 28) โ(x4โ8x3+ 18x2)dx.
ANSWER: 62.5
4. HINT: The average value is 1
5Z5ฯ
0
xsin ๎1
5x๎dx. Use integration by parts to do this integral.
ANSWER: 5
5. ANSWER: Only I is true since the derivative of xeโxis equal to eโx(1 โx). Statement II is
not true since the derivative of eโx(1 โx) is not equal to xeโx.
6. ANSWER: Z1
0
eโx2dx โ0.8220