# Nasty Integration problem

1. Aug 30, 2014

### gaobo9109

1. The problem statement, all variables and given/known data

$\int sin e^{-x}+e^x cos e^{-x}\,dx$

Find the integral above

2. Relevant equations

3. The attempt at a solution

I tried substituting $u=e^{-x}$, but i get $\int \frac{sin u}{u}+\frac{cos u}{u^2} \,du$, which is non-integrable function.

Last edited: Aug 30, 2014
2. Aug 30, 2014

### ehild

What is the derivative of $e^x\cos(e^{-x})$?

ehild

3. Aug 30, 2014

### gaobo9109

Thanks. I didn't notice that

4. Aug 30, 2014

### ehild

You are welcome.

ehild

5. Aug 30, 2014

### nrqed

A second way is to try to transform the cos into a sin function. That leads to the hope that an integration by part will work by starting with

$e^x cos (e^{-x}) = \frac{d e^x}{dx} \cos (e^{-x})$

and sure enough that works.