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How to integrate $$\int \frac{\cos x}{1+a\cos x}\ dx$$ where $a$ is a constant. I've tried substitution, tangent half angle substitution, breaking the fraction apart, and even the typical $u$ $\dfrac{dv}{dx}$ method. Nothing seems to work. Can anyone help?

Nosrati
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koifish
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  • You have asked this a while before. The tangent-half-angle substitution would work. Note that $\sin^2(x) +\cos^2(x) =1 $. – xbh Aug 27 '18 at 04:16
  • Possible duplicate of https://math.stackexchange.com/questions/1740458/finding-int-fracdxab-cos-x-without-weierstrass-substitution?noredirect=1&lq=1 – Nosrati Aug 27 '18 at 04:25

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If you used the tangent half-angle substitution, you ended with $$I=\int\frac{2-2 t^2}{-a t^4+a+\left(t^2+1\right)^2}\,dt$$ Using partial fraction decomposition $$\frac{2-2 t^2}{-a t^4+a+\left(t^2+1\right)^2}=\frac 2a \left(\frac{1}{(a-1) t^2-(a+1)}+\frac{1}{t^2+1}\right)$$ which looks quite simple.

Claude Leibovici
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