Improper Integrals in Mathematics: Convergence and Divergence - Prof. Donna I. Wilson, Study notes of Calculus

Download Improper Integrals in Mathematics: Convergence and Divergence - Prof. Donna I. Wilson and more Study notes Calculus in PDF only on Docsity! Math 1312 Name D.Wilson Improper Integrals 1. If f is continuous on [a,∞), then ∫ ∞ a f(x) dx = lim b→∞ ∫ b a f dx (a) If the limit exists, then the integral converges to the limit. (b) If the limit does not exist, then the integral diverges. 2. If f is continuous on (−∞, b], then ∫ b −∞ f(x) dx = lim a→−∞ ∫ b a f dx (a) If the limit exists, the integral converges to the limit. (b) If the limit does not exist, then the integral diverges. 3. If f is continuous on (−∞,∞), then ∫ ∞ −∞ f(x) dx = ∫ c −∞ f dx + ∫ ∞ c f dx for any real num- ber c. (a) When does ∫ ∞ −∞ f(x) dx converge? (b) When does ∫ ∞ −∞ f(x) dx diverge? 4. Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. (a) ∫ ∞ 0 xe−2x dx (b) ∫ 0 −∞ xe−2x dx (c) ∫ ∞ −∞ 1 3 + 2x2 dx (d) ∫ ∞ 0 ex 1 + ex dx