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Solving the integrals $\int \frac{ \alpha\sin{x} + \beta\cos{x} }{a\sin{x} + b\cos{x}} dx$

I found this as a problem in a book, I solved it, and I kind of liked it. 

So I am going to share the end result here.

The solution to the integral 

$$\int \frac{ \alpha\sin{x} + \beta\cos{x} }{a\sin{x} + b\cos{x}} dx$$ 

is the antiderivative function 

$$F(x) = Ax + B \ln{\left|a\sin{x} + b\cos{x}\right|}$$  


$$A = \frac{\alpha a + \beta b}{a^2+b^2}$$

$$B = \frac{\beta a - \alpha b}{a^2+b^2}$$

One can verify this by differentiating $F(x)$.

The original problem, the one which I encountered was actually asking to find the constants $A$ and $B$.

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