# Homework Help: Integration by substitution

1. Apr 29, 2014

### rwooduk

integral between -infinity and +infinity of

e^-iwt / (a - iw) dw

u = a - iw

du/dw = -i

dw = du / -i

e^-iwt = e^(u+a)t / -iu du

its a homework question, so just a pointer as to which method to use would be appreciated, it says substitution, but ive tried the above and doesnt get me anywhere, maybe im not choosing the correct substitution?

thanks for any help

2. Apr 29, 2014

### Simon Bridge

Have you been instructed to use substitution?
Have you tried ratonalizing the denominator?

3. Apr 29, 2014

### Ray Vickson

$$J \equiv \int \frac{e^{-iwt}}{a-iw} \, dw = i e^{-at} \int \frac{e^{ut}}{u} \, du$$
This last form reveals that the integral is "non-elementary", because $\exp(tu)/u$ cannot be integrated found as a finite expression involving elementary functions; it involves the so-called exponential integral; see http://en.wikipedia.org/wiki/Exponential_integral .