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I think it is necessary to do the replacement. That's just what?

$$\int \frac{3\cos x+7\sin x}{5\sin x+2\cos x} dx$$

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2 Answers 2

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HINT:

Write $$3\cos x+7\sin x=A(5\sin x+2\cos x)+B\dfrac{d(5\sin x+2\cos x)}{dx}$$ where $A,B$ are arbitrary constants.

Compare the coefficients of $\cos x,\sin x$ to determine $A,B$

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Hint: you can substitute $t = tan(\frac{x}{2})$

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