Solution of QUIZ #1 ∫2 1. Decide whether the integral −2 ln(x2 ) dx is improper. Explain your reasoning. solution: ∫ 2 The integral −2 ln(x2 ) dx is improper. Because the function has an infinite discontinuity in [−2, 2] at x = 0. (10%) 2. Determine whether diverges or ∫ ∞ the improper∫ integral ∫ ∞converges. Evaluate the integral if it ∞ converges. (a) 0 e−x/3 dx, (b) e lnxx dx and (c) −∞ x2x+1 dx. (45%) solution: (a) ∫ b lim e−x/3 dx = lim [−3e−x/3 ]b0 = lim −3e−b/3 + 3 = 3(converges) b→∞ (b) b→∞ 0 ∫ b lim b→∞ (c) ∫∞ e b→∞ ln x (ln x)2 b (ln b)2 1 dx = lim [ ]e = lim [ − ] = ∞(diverges) b→∞ b→∞ x 2 2 2 ∫0 ∫∞ dx = −∞ x2x+1 dx + 0 x2x+1 dx(consider the one side) ln(x2 + 1) b ln(b2 + 1) x dx = lim [ ] = lim [ ] = ∞(diverges) 0 b→∞ b→∞ x2 + 1 2 2 x −∞ x2 +1 ∫ b lim b→∞ 0 3. A business is expected to yield a continuous flow of profit at the rate of 5% compounded continuously, what is the present value of the business (a) for 10 years? (b) forever? (20%) solution: (a) ∫1 0f e−0.05t dt = [−20f e−0.05t ]10 0 = −20f e−0.5 + 20f 0 (b) ∫ ∞ −0.05t fe dt = [−20f e−0.05t ]∞ 0 = 20f 0 ∫∞ 4. Show that the improper integral 0 xe−x dx converges, then evaluate the integral. (25%) solution: ∫ b lim b→∞ xe−x dx 0 let u = x, du∫= 1, dv = e−x , v = −e−x lim [xe−x − −e−x dx]b0 = lim [xe−x − e−x ]b0 = lim [(be−b − e−b ) − (−1)] = 1 b→∞ b→∞ b→∞