Improper Integrals

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Mathematics

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Math 133 HW 5 (Section 8.8) (Due Wednesday, October 11 in class) (Please show your work to receive full credits!) Section 8.8 Evaluating improper integrals Determine whether the improper integral converges or diverges. If it converges, find its value. Z 2 1. 1 Z dx x ln x Diverges 1 x ln x dx Converges to the value −1/4 2. 0 Z 3. 3 x dx Converges to the value 9/4 − 1)1/3 0 Hint: the function is discontinuous at x = 1. You must split this integral into the sum of two improper integrals - one from x = 0 to x = 1 and the other from x = 1 to x = 3. Then evaluate each one. (x2 Section 8.8 Comparison tests for improper integrals Determine whether the improper integral converges or diverges. If you use a comparison test, be sure to specify which test you’re using and explain why the hypotheses are satisfied. Z ∞ 1 + cos x dx Converges 1. x2 + 3 1 Z ∞ 3 + sin x 2. dx Diverges x−1 2 Z ∞ 1 3. dx Diverges ln x 2 Z ∞ √ x−1 4. dx Converges 2 x +x+1 1 Z ∞ 2x + 1 √ 5. dx Diverges x3 + 2 1 1
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