<s>[INST] Indefinite Integral with "sin" and "cos": $\int\frac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $ Indefinite Integral with sin/cos
I can't find a good way to integrate: $$\int\dfrac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $$
 [/INST] It is a pity you did not have a minus-sign in the numerator, since
$$
D\ln(3\cos x+2\sin x)=\frac{2\cos x-3\sin x}{3\cos x+2\sin x},
$$
but let us see how we can use this fact anyways.
Let us aim at writing
$$
\frac{2\cos x+3\sin x}{3\cos x+2\sin x}=c_1 \frac{2\cos x-3\sin x}{3\cos x+2\sin x}+c_2 \frac{3\cos x+2\sin x}{3\cos x+2\sin x}
$$
since both those terms are easy to integrate. This leads us to the linear equations $2=2c_1+3c_2$ and $3=-3c_1+2c_2$. The solution to this system is
$c_1=-5/13$ and $c_2=12/13$.
Thus
$$
\int\frac{2\cos x+3\sin x}{3\cos x+2\sin x}\,dx = -\frac{5}{13}\int \frac{2\cos x-3\sin x}{3\cos x+2\sin x}\,dx +\frac{12}{13}\int \frac{3\cos x+2\sin x}{3\cos x+2\sin x}\,dx.
$$
I guess you can take it from here?
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