Evaluate the improper integral whenever it is convergent. If it is divergent, state this.
-$\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } }$ dx
A) 0
B) 1
C) 2
D) divergent
E) -1

Question

Evaluate the improper integral whenever it is convergent.  If it is divergent, state this.  
-$\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } }$ dx
A) 0
B) 1
C) 2
D) divergent
E) -1

Quizplus · Accepted Answer

The integral of $xe^{-x^2}$ from $-\infty$ to $\infty$ is a classic example of an odd function being integrated over a symmetric interval. The function $f(x) = xe^{-x^2}$ is odd because $f(-x) = -f(x)$, and when you integrate an odd function over a symmetric interval around zero, the result is always 0. This is because the negative side cancels out the positive side exactly.

Evaluate the Improper Integral Whenever It Is Convergent ∫−∞∞xe−x2\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } }∫−∞∞​xe−x2

Evaluate the Improper Integral Whenever It Is Convergent $\int _ { - \infty } ^ { \infty } x e ^ { - x ^ { 2 } }$