8x − 4 x2(x2+4)dx

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) = xe^x. (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

te^tdt 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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