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