Evaluate the improper integral if it converges, or state that it diverges.
E) diverges

Question

Evaluate the improper integral if it converges, or state that it diverges. ​  ​
E) diverges

Quizplus · Accepted Answer

To evaluate the improper integral, we can first try to use the limit comparison test to compare it to a known divergent integral. Let's choose the function f(x) = 1/x.

Then, we have:

lim x→∞ (x^2)/(x(1 + x^2)) = lim x→∞ 1/(1 + 1/x^2) = 1

lim x→∞ 1/x = 0

Since the limit of the quotient is nonzero and finite, and the limit of the denominator is zero, we cannot use the limit comparison test to determine convergence.

Alternatively, we can try to use partial fractions to simplify the integrand:

(x^2)/(x(1 + x^2)) = 1/x - 1/(1 + x^2)

However, the first term 1/x is a known divergent integral. Therefore, the entire integral must also diverge.

Thus, the best choice is E, which states that the integral diverges.

Evaluate the Improper Integral If It Converges, or State That