MATH 1120 Midterm: MATH 1120 Cornell Current e2sol4 30 18
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Problem 1 (20pts): use integration, the direct comparison test, or the limit comparison test to test whether the following integrals converge. If they converge, compute their values. (a) z e2. By the chain rule we have du = 1 x dx. Also, the lower ln x limit for integration is ln(ln(e2)) = ln 2, while the upper limit is . Then the improper integral can be rewritten as. Therefore, we can conclude that the improper integral diverges. lim b z b ln 2. 1 u du = lim b ln(b) ln(ln 2) = . (b) z 1. The integrand is continuous throughout the interval [0, 1]. To evaluate, we can apply the trigonometric substitution x = tan . Also, the lower and upper limits for integration are 0 and . Then the integral can be computed as follows: 0 x ln x dx (c) z 1. We rst observe that as x 0, ln x .
