Find the Fourier coefficients and Fourier series of the following function

Question

Find the Fourier coefficients and Fourier series of the following function $$f(x)=\frac{\pi e^{-x}}{e^{\pi}-e^{-\pi}};\quad -\pi\le x\le\pi$$

Here is my work:

Since $L=\pi$ we obtain \begin{align*}a_0&=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(x)\text{d}x=\frac{1}{2\left(e^{\pi}-e^{-\pi}\right)}\int\limits_{-\pi}^{\pi}e^{-x}\text{d}x=\frac{1}{2\left(e^{\pi}-e^{-\pi}\right)}\Big[-e^{-x}\Big]_{-\pi}^{\pi}=\frac{1}{2} \\ a_n&=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)\cos\left(nx\right)\text{d}x=\frac{1}{e^{\pi}-e^{-\pi}}\int\limits_{-\pi}^{\pi}e^{-x}\cos(nx)\text{d}x \\ b_n&=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)\sin(nx)\text{d}x=\frac{1}{e^{\pi}-e^{-\pi}}\int\limits_{-\pi}^{\pi}e^{-x}\sin(nx)\text{d}x\end{align*}

I'm stuck in $a_{n}$ and $b_{n}$, need help finding an easy way to solve this integration.

Answer 1

Hint: $$\int e^{-x + i a x} dx = \int{e^{-x} \cos(ax)dx + i \int e^{-x} \sin(ax) dx}.$$

Answer 2

IBP does the trick. Here's a start:

$$ \frac{1}{n}\int_{-\pi} ^\pi e^{-x} \cos{nx} \ dx = \frac{1}{n} \left(-e^{-x} \cos {nx}\Big|_{-\pi}^\pi - n\int_{-\pi} ^\pi e^{-x} \sin{nx} \ dx \right) $$

The first term is easy. Do another IBP on the second term and you'll get the original integrand, so do what you would do in calculus II and solve for the unknown integral afterwards.