How can the integral of a complex function be simplified?

[gaobo9109]

Homework Statement

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

Find the integral above

Homework Equations

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.

 

[ehild]

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

ehild

 

[gaobo9109]

Thanks. I didn't notice that

 

[ehild]

You are welcome.

ehild

 

[nrqed]

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

ehild

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.