Calculating $\int_{\pi}^{\infty} \cfrac{\cos(x+t)}{t} dt$

Question

I'm having trouble calculating the integral: $$\int_{\pi}^{\infty} \cfrac{\cos(x+t)}{t} dt$$ Can anyone help me with this I have no clue what to do.

take a look the defintions for cosine and sine integrals. https://en.wikipedia.org/wiki/Trigonometric_integral — Math-fun, Nov 01 '17 at 10:45
Are you supposed to evaluate it or just prove con/divergence? — Randall, Nov 01 '17 at 11:45

score 1 · Accepted Answer · answered Nov 01 '17 at 10:46

This involves special functions.

$$\cos(x+t)=\cos(x)\cos(t)-\sin(x)\sin(t)$$ makes $$\int \cfrac{\cos(x+t)}{t}\, dt=\cos(x)\int \cfrac{\cos(t)}{t} dt-\sin(x)\int \cfrac{\sin(t)}{t} dt=\text{Ci}(t) \cos (x)-\text{Si}(t) \sin (x)$$ where appear the sine and cosine integrals.

Calculating $\int_{\pi}^{\infty} \cfrac{\cos(x+t)}{t} dt$

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