Solve the improper integral $ b\int_{-\infty}^0 xe^x dx $

Question

$ b\int_{-\infty}^0 xe^x dx $ my solution gets me to an undefined limit $0 \cdot (-\infty) $ while the answer is supposed to be $-b$

my solution: $ b\cdot\int xe^x dx = b \cdot e^x(x-1) = b \cdot ((e^0 \cdot (0 -1)) - (e^{-\infty}\cdot(-\infty - 1)) $ the first part equals to $-1$. The second is $-(0 \cdot(-\infty - 1))$, which is undefined limit. Am I wrong? Thanks for help.

Answer 1

Change the integration variable from $x$ to $-x$ and you will get $$ \int_\infty^0(-x)e^{-x}(-dx)=-\int_0^\infty xe^{-x}dx. $$ Then, integrating by parts you will get $$ \left.xe^{-x}\right|_0^\infty-\int_0^\infty e^{-x}dx=-1. $$ Use has been made of the fundamental limit $\lim_{x\rightarrow\infty}xe^{-x}=0$.

Answer 2

Just by Hospital: $$\lim_{x\rightarrow-\infty}xe^x=\lim_{x\rightarrow-\infty}\frac{x}{e^{-x}}=\lim_{x\rightarrow-\infty}\frac{1}{-e^{-x}}=0$$ and also,$$\lim_{x\rightarrow-\infty}e^x=0.$$