Evaluate the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$
A)0
B)1
C) $e$
D) $e ^ { - 1 }$

E) $e - 1$
F) $\frac { 1 } { 2 }$
G)2
H)Divergent

Question

Evaluate the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$ 
A)0
B)1
C) $e$ 
D) $e ^ { - 1 }$

E) $e - 1$ 
F) $\frac { 1 } { 2 }$ 
G)2
H)Divergent

Quizplus · Accepted Answer

The integral can be solved using substitution and recognizing it as a form of the Gamma function. Let $ u = x^2 $, then $ du = 2x \, dx $. The integral becomes $\frac{1}{2} \int_0^\infty e^{-u} \, du$, which evaluates to $\frac{1}{2}$.

Evaluate the Improper Integral ∫0∞xe−x2dx\int _ { 0 } ^ { \infty } x e ^ { - x ^ { 2 } } d x∫0∞​xe−x2dx

Evaluate the Improper Integral $\int _ { 0 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$