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## Integrals of Rational Functions times Trig function

Moderator: galactus

### Integrals of Rational Functions times Trig function

Mon Sep 30, 2013 5:06 pm

Posts: 850
Location: Jaipur, India

I am trying to find closed forms for integrals of rational functions times a trigonometric function.

I obtained some totally astonishing results like

$\displaystyle \int_0^\infty \frac{\cos x}{5-4\cos x} \times \frac{1}{1+x^2}dx = \frac{\pi }{2e-1}\left[ \frac{e^2-1}{e-2}\left\{ \frac{2}{3}-\frac{e}{e^2-1}\right\}-\frac{e}{2} \right]=0.5568468705 \cdots$

Mathematica can't evaluate this integral.

Most of my results are recorded here.

### Re: Integrals of Rational Functions times Trig function

Mon Sep 30, 2013 7:02 pm
zaidalyafey
Global Moderator

Posts: 354
These seem interesting results , keep going .
$\displaystyle \sum_{k\geq 0}\frac{f^{(k)} (s)}{(s+1)_k} (-s)^{k}= \int^{1}_0x^{s} f(s x) \left( \frac{x}{s}\right) \, dx$

Wanna learn what we discuss , see tutorials