1. Compute the integrals.
(a) (6%) ∫ x(sin 2x + cos x)dx. (b) (9%) ∫
8x − 4 x2(x2+4)dx.
2. (a) (3%) Evaluate and simplify d dxln (
√
x2+1 + x).
(b) (5%) Evaluate ∫ sec x dx.
(c) (7%) Use (b) and trigonometric substitution to find ∫
1 0
1
√
x2+1dx.
(d) (7%) Use (a) and integration by parts to evaluate the integral ∫
1 0 ln (
√
x2+1 + x) dx.
3. (a) (10%) Let R be the region bounded by y = cos x, y = sin 2x and x = 0 in the first quadrant.
Rotate R about the x-axis. Find the volume of the resulting solid.
(b) (6%) (A) ∫
1 0
πx dx. (B) ∫
1 0
πx4dx. (C) ∫
1 0
πx2dx. (D)∫
1 0
x2dx.
Match each solid with the integral that represents its volume.
Integrals A B C D Solid
4. (8%) Find the length of the curve
y = f (x) =∫
x 1
√
t3−1 dt, 1 ≤ x ≤ 4.
5. Let f (x) = xex. (When finding the following Taylor series, you don’t need to specify the range of x for which the series equals the function.)
(a) (4%) Find the Taylor series for f (x) at x = 0.
(b) (7%) Calculate ∫
x 0
tetdt and find its Taylor series at x = 0.
(c) (4%) Find the sum
∞
∑
n=0
1 n!(n + 2).
6. (a) (6%) Find the Taylor series for f (x) = ln(1 − x2), g(x) = cos x − 1, and h(x) = sin(2x4)at x = 0.
(You don’t need to specify the range of x for which the function equals its Taylor series.)
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(b) (4%) Evaluate lim
x→0
(cos x − 1) ln(1 − x2) sin(2x4)
.
7. (a) (6%) Compute lim
x→0
∫
4x2
0 cos(√ t)dt
x2 by L’Hospital’s Rule.
(b) (8%) Compute lim
x→∞(1 + x)1/ ln x.
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