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Integrals of Rational Functions of Sine and Cosine


Definition

The integrals of rational functions of sine and cosine are integrals where the integrand is a rational function composed of sines and cosines, usually in the denominator, which is not easily reducible. However, these integrals are easy to solve by applying the recommended substitution: $z=\tan\left(\frac{x}{2}\right)$

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Problems Solved

Find the integral $\int\left(\frac{1}{2+\sin\left(x\right)+\cos\left(x\right)}\right)dx$
Evaluate the integral $\int_0^{\pi}\frac{1}{1+2sen^2x}dx$
$\frac{dy}{dx}=-\frac{e^{-y^2}}{y\left(2\sin^2\left(x\right)-2\right)}$
Find the integral $\int\frac{1}{\sin\left(x\right)-\cos\left(x\right)}dx$
Find the integral $\int\frac{1}{3+2cos\left(x\right)+2sin\left(x\right)}dx$
Evaluate the integral $\int_0^{\pi}\left(\frac{1}{1-\cos x}\right)dx$
Find the integral $\int\frac{dz}{1-\sin\left(z\right)}$

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