1. (12%) Suppose that a continuous function f (x) satisfies the equation ∫
x2 a
f (√ t)
t dt + 2 = x, for all x > 0. Find f (x) for x > 0 and the constant a > 0.
2. (24%) Evaluate the integrals.
(a) (8%) ∫
1
0 sin−1x dx. (b) (8%) ∫ tan4x dx. (c) (8%) ∫
1 x4−1dx.
3. (12%) Evaluate the improper integral ∫
∞ 0
e−xsin x dx.
4. (8%) Find the arc length of the curve y = e2x+e−2x
4 from x = 0 to x = 1.
5. (12%) Compute the volume of the solid obtained by rotating the disc, x2+ (y − 2)2≤1, about the x-axis.
6. (16%)
(a) (8%) Write down the Taylor expansion of (1 − x2)−
1
2 at x = 0.
(b) (8%) Derive from a) the Taylor expansion of sin−1x at x = 0 and write down the first three nonzero terms.
7. (16%) Compute the following limits.
(a) (8%) lim
x→0
sin(x2)
ex−1 − x. (b) (8%) lim
x→∞(1 +√ x)ln x1 .
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