Fourier series for e^|x| on (-1,1) with period 2

[anoncow]

Homework Statement

$\ f(x) = e^{|x|}$ with $x \in (-1,1)$ and f(x+2) = f(x) $\forall x$

Homework Equations

The Attempt at a Solution

Linked solution

What am I meant to do once I get to the last line? (assuming all is right up until then)

 

[LCKurtz]

Here is that last line.

$$ 2\int_0^1 e^x \cos(n \pi x)\,dx =
\frac{2}{n \pi} \left[e^x\sin(n \pi x) \right]_0^1 -
\frac{2}{n^2 \pi^2} \left[e^x\cos(n \pi x) \right]_0^1 -
\frac{2}{n^2 \pi^2} \int_0^1 e^x \cos(n \pi x)$$

Treat that integral as an unknown and solve for it and, of course, put in the evaluated limits.