Evaluate $\int\limits_0^{\infty}e^{-x}|\sin (\frac{π}{4}-x)|dx$

Question

I have to evaluate the integral :

$$\int\limits_0^{\infty}e^{-x}|\sin (\frac{π}{4}-x)|dx$$

By wolfram alpha :

$$\int\limits_0^{\infty}e^{-x}|\sin (\frac{π}{4}-x)|dx≈0,476531..$$

$$\sin (\frac{π}{4}-x)>0\implies x\in ]πn-\frac{7π}{4},πn-\frac{3π}{4}[,n\in\mathbb{Z}$$

$$\sin (\frac{π}{4}-x)<0\implies x\in ]2πn+\frac{π}{4},2πn+\frac{5π}{4}[,n\in\mathbb{Z}$$

So I can't split integral

Normally I would show my work so far, however, I don't even know where to start with this question!

Answer 1

$$\begin{align} I&=\int_0^\infty e^{-x}\left|\sin\left(\frac\pi4-x\right)\right|\mathrm{d}x\\ &=\int_0^\infty e^{-x}\left|\sin\left(x-\frac\pi4\right)\right|\mathrm{d}x\\ &=\int_{-\pi/4}^\infty e^{\pi/4-u}\left|\sin{u}\right|\mathrm{d}u\\ &=e^{\pi/4}\int_{-\pi/4}^\pi e^{-u}\left|\sin{u}\right|\mathrm{d}u+\sum_{n=1}^\infty e^{\pi/4}\int_{n\pi}^{(n+1)\pi} e^{-u}\left|\sin{u}\right|\mathrm{d}u\\ &=e^{\pi/4}\int_{-\pi/4}^\pi e^{-u}\left|\sin{u}\right|\mathrm{d}u+\sum_{n=1}^\infty e^{\pi/4}\left|\int_{n\pi}^{(n+1)\pi} e^{-u}\sin{u}\,\mathrm{d}u\right| \end{align}\\ $$

Can you finish it?