Evaluate the improper integral $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x$
A) $e$

B) $e ^ { 2 } - 1$
C)0
D) $e ^ { 2 }$
E) $e ^ { - 1 }$
F) $e ^ { - 2 }$
G)1
H) $1 - e ^ { - 2 }$

Question

Evaluate the improper integral $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x$
A) $e$

B) $e ^ { 2 } - 1$ 
C)0
D) $e ^ { 2 }$ 
E) $e ^ { - 1 }$ 
F) $e ^ { - 2 }$ 
G)1
H) $1 - e ^ { - 2 }$

Quizplus · Accepted Answer

This integral evaluates to 0 because the function $x e^{-x^2}$ is an odd function (i.e., $f(-x) = -f(x)$), and the integral is over a symmetric interval around 0. The area under the curve from $-\infty$ to 0 is the negative of the area from 0 to $\infty$, so they cancel each other out.

Evaluate the Improper Integral ∫−∞∞xe−x2dx\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x∫−∞∞​xe−x2dx

Evaluate the Improper Integral $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } } d x$