$\text { Determine whether the improper integral } \int _ { 9 } ^ { 11 } \frac { 2 } { ( x - 10 ) ^ { 2 } } d x \text { diverges or converges. }$ Evaluate the integral if it converges.
A) 100
B) 2
C) $\frac { 1 } { 50 }$
D) $\frac { 22 } { 9 }$
E) diverges

Question

Quizplus · Accepted Answer

The integral $\int _ { 9 } ^ { 11 } \frac { 2 } { ( x - 10 ) ^ { 2 } } d x$ is improper because the integrand becomes infinite at $x = 10$. To determine convergence, we split the integral at the point of discontinuity and evaluate the limit as we approach the problematic point from both sides. However, due to the nature of the function $ \frac { 2 } { ( x - 10 ) ^ { 2 } }$, which has a vertical asymptote at $x = 10$, the integral diverges because the area under the curve approaches infinity as $x$ approaches 10.