Today I had an exam and I mixed up the integration by parts formula.
The question was to integrate
$$ \int\nolimits_0^\infty \frac{e^{-at} – e^{-bt}}{t} \text{d}t $$
I will try solve this again with the right formula when I arrive home. I would appreciate if somebody could tell me the solution so I can double check and maybe give a hint to another way of solving this instead of integration by parts (if possible).
Best Answer
I may as well give an answer:
Note $$ \frac{ e^{-at} - e^{-bt} }{t} = \int^b_a e^{-xt} dx $$ so our integral is
$$ \int^{\infty}_0 \int^b_a e^{-xt} dx dt = \int^b_a \int^{\infty}_0 e^{-xt} dt dx $$ $$ = \int^b_a \frac{1}{x} dx = \log(b/a) $$
This is a general method, and often this whole process is compressed into a well known integral called Frullani's Integral.