Changing Bounds and Multiply by -1

Question

Sorry my calculus knowledge is extremely rusty - what is the reason that we can flip the bounds from $-\infty\to 0$ to $0\to\infty$ and then also flip the $x$ value to $-x$?

Answer 1

It's easier to see if you use another variable for the substitution, say $y=-x$. In the integral $\int_{-\infty}^0 e^{\epsilon x} f_X(x)dx$, you need to:

• replace $x$ with $-y$ in the integrand;
• change the bounds from $(-\infty,0)$ to $(\infty, 0)$, because $y\rightarrow \infty$ when $x \rightarrow \infty$, and $y=0$ when $x=0$;
• set $dx=-dy$, because $\frac{dy}{dx}=-1$.

Putting everything together gives $$ \int_{-\infty}^0 e^{\epsilon x} f_X(x)dx = -\int_{\infty}^0 e^{-\epsilon y} f_X(-y)dy = \int_0^{\infty} e^{-\epsilon u} f_X(-y)dy $$

Also, the very last integral in your expression should be $\int_{-\infty}^\infty e^{-\epsilon x}f_X(-x)dx$.