Determine whether the improper integral converges or diverges.
-$$\int _ { 2 } ^ { \infty } \frac { 4 } { ( x + 1 ) ^ { 2 } } d x$$
A) Diverges
B) Converges

Question

Quizplus · Accepted Answer

The integral can be rewritten as
$$\int_2^\infty \frac{4}{(x+1)^2}dx=\int_3^\infty\frac{4}{u^2}du$$
where $u=x+1$. The integral $\int_3^\infty\frac{4}{u^2}du$ is a $p$-integral with $p=2>1$, so it converges. Therefore, the original integral also converges.

Determine Whether the Improper Integral Converges or Diverges ∫2∞4(x+1)2dx\int _ { 2 } ^ { \infty } \frac { 4 } { ( x + 1 ) ^ { 2 } } d x∫2∞​(x+1)24​dx

Determine Whether the Improper Integral Converges or Diverges $\int _ { 2 } ^ { \infty } \frac { 4 } { ( x + 1 ) ^ { 2 } } d x$