Evaluate the improper integral or state that it is divergent.
-$$\int _ { 6 } ^ { \infty } \frac { d x } { x ^ { 2 } - 25 }$$
A) $\frac { 1 } { 10 } \ln \frac { 1 } { 6 }$
B) $- \frac { 1 } { 5 } \ln 11$
C) $\frac { 1 } { 10 } \ln 11$
D) $\frac { 1 } { 10 } \ln 6$

Question

Quizplus · Accepted Answer

We can use partial fractions to write the integrand as
$$\frac{1}{x^2-25}=\frac{1}{10}\left(\frac{1}{x-5}-\frac{1}{x+5}\right)$$
Then the integral becomes
$$\int_6^\infty\frac{dx}{x^2-25}=\frac{1}{10}\left(\int_6^\infty\frac{dx}{x-5}-\int_6^\infty\frac{dx}{x+5}\right)$$
Since both integrals on the right-hand side diverge, their difference converges by the limit comparison test. Specifically, we have
$$\lim_{t\to\infty}\frac{\int_6^t\frac{dx}{x-5}}{\ln t}=-\frac{1}{5},\ \ \ \ \ \ \ \ \lim_{t\to\infty}\frac{\int_6^t\frac{dx}{x+5}}{\ln t}=\frac{1}{5}$$
Therefore, the integral converges to
$$\frac{1}{10}\left(-\frac{1}{5}+\frac{1}{5}\right)=\boxed{\frac{1}{10}\ln 11}$$

Evaluate the Improper Integral or State That It Is Divergent ∫6∞dxx2−25\int _ { 6 } ^ { \infty } \frac { d x } { x ^ { 2 } - 25 }∫6∞​x2−25dx​

Evaluate the Improper Integral or State That It Is Divergent $\int _ { 6 } ^ { \infty } \frac { d x } { x ^ { 2 } - 25 }$