# Practice Final Solutions - 1 Mathematics 19B Winter 2002 V...

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1 Mathematics 19B; Winter 2002; V. Ginzburg Practice Final, Solutions 1. For each of the ten questions below, state whether the assertion is true or false : (a) To evaluate a 2 - x 2 dx one should use the trigonometric substitution x = a sin θ with - π/ 2 θ π/ (b) sinh x dx = - cosh x + C (c) If a n 0, then the series n =1 a n converges. (d) 1 (e) To find the integral x 3 e x dx one should apply the method of integration (f) Assume that lim n →∞ n | a n | = L exists and L 1. Then the series n =1 a n (g) Assume that b n is a decreasing sequence, b n > 0 for all n , and lim n →∞ b n = 0. Then the series n =1 ( - 1) n b n (j) Let a n be a bounded monotonic sequence. Then a n converges. 2. Evaluate the following indefinite integrals: (a) x x 2 - 5 x +6 dx .
2 Solving for A and B , we obtain A = 3 and B = - 2. Hence, x x 2 - 5 x + 6 = 3 x - 3 - 2 x - 2 and x x 2 - 5 x + 6 dx = 3 x - 3 - 2 x - 2 dx = 3 ln | x - 3 | - 2 ln | x - 2 | + C.