Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 0 } ^ { + \infty } x ^ { 2 } e ^ { - 10 x } \mathrm {~d} x$
A) $- \frac { 1 } { 500 }$
B) $\frac { 1 } { 10 }$
C) $\frac { 1 } { 500 }$
D) 0
E) diverges

Question

Decide whether the integral converges. If the integral converges, compute its value.   $\int _ { 0 } ^ { + \infty } x ^ { 2 } e ^ { - 10 x } \mathrm {~d} x$  
A) $- \frac { 1 } { 500 }$
B) $\frac { 1 } { 10 }$
C) $\frac { 1 } { 500 }$
D) 0
E) diverges

Quizplus · Accepted Answer

This integral can be solved using integration by parts or recognizing it as a gamma function. The integral of $x^2e^{-10x}$ from 0 to $\infty$ converges because $e^{-10x}$ decays to 0 much faster than $x^2$ grows. The exact value can be found by integrating by parts twice or using the formula for the gamma function, resulting in $\frac{2}{10^3} = \frac{1}{500}$.

Decide Whether the Integral Converges ∫0+∞x2e−10x dx\int _ { 0 } ^ { + \infty } x ^ { 2 } e ^ { - 10 x } \mathrm {~d} x∫0+∞​x2e−10x dx

Decide Whether the Integral Converges $\int _ { 0 } ^ { + \infty } x ^ { 2 } e ^ { - 10 x } \mathrm {~d} x$