Find the value of the improper integral that converges.
\[
\int_{-\infty}^{\infty} x e^{-4 x^{2}} d x
\]
A. -4
B. $\frac{1}{e^{4}}$
C. 0
D. The improper integral diverges.
Answer
Final Answer: \(\boxed{0}\)
Step 1 ：Given the integral \(\int_{-\infty}^{\infty} x e^{-4 x^{2}} d x\)
Step 2 ：Notice that the function \(x e^{-4 x^{2}}\) is an odd function, which means it is symmetric with respect to the origin.
Step 3 ：The integral of an odd function from negative infinity to positive infinity is always zero. This is because the areas under the curve on the positive and negative sides of the y-axis are equal in magnitude but opposite in sign, so they cancel each other out.
Step 4 ：Thus, the value of the integral \(\int_{-\infty}^{\infty} x e^{-4 x^{2}} d x\) is 0.
Step 5 ：Final Answer: \(\boxed{0}\)
