Calculus II, Winter 09
Grinshpan
FINAL EXAM ANSWERS
1 A)∫ 2
11x3 dx = 3
8
1 B)∫ ln 3
0ex√
1 + ex dx = 43(4−
√2)
2 A)∫t3 ln t dt = 1
4t4 ln t− 1
16t4 + C
2 B)∫
arctanx dx = x arctanx− 12
ln(x2 + 1) + C
3 A)∫
sin4 θ cos5 θ dθ = 15
sin5 θ − 27
sin7 θ + 19
sin9 θ + C
3 B)∫
dxx2+x−2
= 13
ln∣∣x−1x+2
∣∣+ C
4 A) π∫ 1
0(t2 − t4)dt = 2π
15(t = x− 1)
4 B) π∫ 1
0((1 +
√y)2 − (1 + y)2)dy = π
2
5 A) y = 12(x2+1)
is the solution of y′ = −4xy2, y(0) = 12
5 B) 25πρ∫ 6
0(9− h)dh = 900πρ
6 A) 4-petal rose6 B) dy
dx
∣∣θ=π
6
= tan(2θ)∣∣θ=π
6
=√
3
7 A)∫∞
0
√5e−4θ dθ =
√5
2
7 B) 2 · 12
∫ π2
π6
((3 sin θ)2 − (1 + sin θ)2)dθ = π
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