Math 1B: Quiz 2 ANSWERS
GSI: Theo Johnson-Freyd Monday, 29 June 2009
You must always justify your answers. This means: show your work, show it neatly, and when
in doubt, use words (and pictures!) to explain your reasoning. No justification = no points.
1. (5 pts) Evaluate the integral Zx−1
x2+ 2xdx.
We decompose the integrand into partial fractions:
x2+ 2x=x(x+ 2) factoring
x−1
x2+ 2x=A
x+B
x+ 2 partial fraction ansatz
x−1 = A(x+ 2) + Bx cross-multiply
0−1 = A(0 + 2) + B0 when x= 0
−2−1 = A(−2 + 2) + B(−2) when x=−2
x−1
x2+ 2x=−1/2
x+3/2
x+ 2 solving for Aand Band substituting
Thus, we can evaluate the integral, using the fact that R1
x+adx = ln |x+a|+C:
Zx−1
x2+ 2xdx =Z−1/2
x+3/2
x+ 2dx
=−1
2ln |x|+3
2ln |x+ 2|+C
= ln s
(x+ 2)3
x
+C
As with Rdx/x, the value of the constant Cmay change at any point of discontinuity; in this
case, the value of Cmay change at x=−2 and at x= 0.
2. (5 pts) Evaluate the integral Zx2
(4 −x2)3/2dx.
We substitute x= 2 sin θ, restricting ourselves to −π/2< θ < π/2 (since |x| ≥ 2 is outside
the domain of the integrand). Then dx = 2 cos θ dθ and:
Zx2
(4 −x2)3/2dx =Z(2 sin θ)2
(4 −4 sin2θ)3/22 cos θ dθ
=Z8 sin2θcos θ
8 cos3θdθ
=Ztan2θ dθ
=Zsec2θ−1dθ
= tan θ−θ+C
=x
√4−x2−arcsin x
2+C