Evaluate the improper integral or state that it is divergent.
-$$\int _ { 0 } ^ { \infty } 21 x e ^ { 2 x } d x$$
A) 1.3333
B) Divergent
C) 1.6667
D) 2.6667

Question

Quizplus · Accepted Answer

This integral is divergent. We can see this by using the limit comparison test with the divergent integral $\int_0^\infty xe^{2x} dx$. Let $f(x) = 21xe^{2x}$ and $g(x) = xe^{2x}$. Then, $\lim_{x	o\infty} \frac{f(x)}{g(x)} = \lim_{x	o\infty} 21 = 21 >0$. Since $\int_0^\infty g(x) dx$ is divergent, then so is $\int_0^\infty f(x)dx$ by the limit comparison test.

Evaluate the Improper Integral or State That It Is Divergent ∫0∞21xe2xdx\int _ { 0 } ^ { \infty } 21 x e ^ { 2 x } d x∫0∞​21xe2xdx

Evaluate the Improper Integral or State That It Is Divergent $\int _ { 0 } ^ { \infty } 21 x e ^ { 2 x } d x$