Evaluate the following improper integral whenever it is convergent.
A) 4
B) -1
C) 1
D) 7
E) the integral is divergent

Question

Evaluate the following improper integral whenever it is convergent. 
A) 4 
B) -1 
C) 1 
D) 7 
E) the integral is divergent

Quizplus · Accepted Answer

The integral is given as 
∫_1^∞▒〖(lnx)/(x^2 ) dx〗 
We can apply the limit comparison test by dividing the function by 1/x^2. Thus, let f(x) = (ln x) / x^2 and g(x) = 1/x^2. Then,

lim (x → ∞) f(x) / g(x) = lim (x → ∞) [(ln x) / x^2] / [1/x^2] 
= lim (x → ∞) ln x 
= ∞

Since the limit is infinite, the integral diverges. Therefore, the answer is E.

Evaluate the Following Improper Integral Whenever It Is Convergent