Quiz 8.8 Part I:
Question:
Evaluate \( \int \limits _{-\infty} ^\infty 2xe^{-x^2} dx \)
Solution:
\( =\lim \limits _{a \rightarrow -\infty} \int \limits _a ^0 2xe^{-x^2}dx + \lim \limits _{b \rightarrow \infty} \int \limits _0 ^b 2xe^{-x^2}dx \)
\( =\lim \limits _{a \rightarrow -\infty} -\left ( 1 - e^{-a^2} \right ) + \lim \limits _{b \rightarrow \infty} - \left ( e^{-b^2} -1 \right ) \)
\( = -1 +1 = 0 \)
\( u = -x^2 \)
\( du=-2xdx \)
\( =\lim \limits _{a \rightarrow -\infty} - \int \limits _{-a^2} ^0 e^u du + \lim \limits _{b \rightarrow \infty} - \int \limits _0 ^{-b^2} e^v dv \)\( du=-2xdx \)
\( =\lim \limits _{a \rightarrow -\infty} -\left ( 1 - e^{-a^2} \right ) + \lim \limits _{b \rightarrow \infty} - \left ( e^{-b^2} -1 \right ) \)
\( = -1 +1 = 0 \)
