Determine whether the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x$ diverges or converges. Evaluate the integral if it converges.
A) 9
B) $e ^ { - 1 / 3 }$
C) $- \frac { 1 } { 3 }$
D) 3
E) diverges

Question

Determine whether the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x$ diverges or converges. Evaluate the integral if it converges. 
A) 9
B) $e ^ { - 1 / 3 }$
C) $- \frac { 1 } { 3 }$
D) 3
E) diverges

Quizplus · Accepted Answer

The integral converges because the function $x e^{-x/3}$ decays exponentially as $x \to \infty$. Using integration by parts, the integral evaluates to 9.

Determine Whether the Improper Integral ∫0∞xe−x/3dx\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x∫0∞​xe−x/3dx

Determine Whether the Improper Integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 3 } d x$