which should be my bounds of integration be after the following substitution?

Question

Okay , Given the following integral

$$\int_0^\infty \frac {e^u}{u} du $$

I let $u = \ln(t+1)$

du = $\frac{1}{1+t} dt $

which would be my new bounds of integration? namely

$$\int_a^b$$

Note- i am not interested in solving the integral ,just interested in understanding when and how to change the limits of integration in a case such as this.

Thank you very much for your help and time!

Answer 1

If you are considering the integral $$\int_a^b f(x)dx$$ and you wish to make the substitution $x=g(t)$, where $g$ is bijective on $(a,b)$, then the substitution results in the integral $$\int_{g^{-1}(a)}^{g^{-1}(b)} f(g(x))g'(x)dx$$ with bounds $g^{-1}(a)$ and $g^{-1}(b)$.