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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

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

where

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

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

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 $B$.