# Taylor's series of $f(x) = \ \int_{\frac{\pi}{2}}^{x} \frac{\cos(t)}{t - \frac{\pi}{2}}dt$

Be the function $$f(x) = \ \int_{\frac{\pi}{2}}^{x} \frac{\cos(t)}{t - \frac{\pi}{2}}dt.$$ I tried to find the Taylor's series of $f(x)$, but I didn't succeed. What is the Taylor series of $f(x)$? Is there a clever way to find it?

Please let me know if the question is unclear.

Hint: $\cos t=\sin(\pi/2-t)$. Change the variable and then integrate the Maclaurin's formula for $\frac{\sin x}{x}$.
• @J.Doe Power series can be differentiated and integrated within their radius of convergence (because the convergence is uniform there). Here the radius is infinity (because of $\sin$). – A.Γ. Oct 13 '17 at 14:42