How to calculate the indefinite integral of $f(x)=\frac{1}{A+B \cos 2x +C \cos x}$.

Question

I have tried substituting $x = \tan t$ but not able to get an integrable expression.

Answer 1

Hint: First use

$$\cos(x)=\dfrac{\exp(ix)+\exp(-ix)}{2}$$ $$\cos(2x)=\dfrac{\exp(2ix)+\exp(-2ix)}{2}$$

then substitute

$$\exp(ix)=u \implies i\exp(ix)dx = du \implies dx = \dfrac{du}{iu}.$$

The resulting integral can be solved by partial fractions.

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A standard alternative approach is to use (advantage is that you avoid complex numbers; disadvantage is that you have to remember this substitution :D)

$$t = \tan \dfrac{x}{2}$$ and substitute $\cos x = \dfrac{1-t^2}{1+t^2}$ and $dx = \dfrac{2dt}{1+t^2}$.