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1.28 Reducing sine and cosine integrals

Theorem.

Let R(cosx,sinx)R(\cos x,\sin x) be an expression that arises from cosx\cos x, sinx\sin x and real constants by addition, multiplication and division. Then

R(cosx,sinx)dx\int R(\cos x,\sin x)dx

reduces to the integral of a rational function by the substitution t=tanx2,t=\tan\frac{x}{2}, hence can be integrated by the method of partial fractions.