Question

Improper Integrals. Say that f is integrable on [a,[infinity]) if limb→[infinity]

R b a f(x)dx =: R [infinity] a f(x)dx exists.
a. For which real values of p does R [infinity] e (logx)p x dx exist?
b. Show that R [infinity] 0 sinx x dx exists. HINT: Consider the alternating series ∑n≥0 R (n+1)π nπ sinx x dx

Answer 1

a. For the improper integral R [infinity] e (logx)p x dx to exist, p must be less than zero. b. The integral R [infinity] 0 sinx x dx exists by using the Alternating Series Test.

Step-by-step explanation:

a. To determine the values of p for which the improper integral R [infinity] e (logx)p x dx exists, we need to consider the convergence of the integral. For the integral to converge, the function inside the integral must approach zero as x approaches infinity. In this case, we have e (logx)p x dx, and since logx approaches infinity as x approaches infinity, we want the exponent p to be less than zero in order for the function to approach zero. Therefore, the values of p for which the integral exists are p < 0.

b. To show that R [infinity] 0 sinx x dx exists, we can use the alternating series ∑n≥0 R (n+1)π nπ sinx x dx. This series converges for all values of x, including infinity. Therefore, by the Alternating Series Test, the integral R [infinity] 0 sinx x dx exists.