# Find the indefinite integral $\int \frac{25}{(3\cos(x)+4\sin(x))^2} dx$

Find the indefinite integral: $$\int\frac{25}{(3\cos(x)+4\sin(x))^2} dx$$

Obviously the first thing to is to expand the denominator but that doesn't help us that much. I have also tried applying the Weierstrass substitution but that lead to a black hole of algebra. I also notice the denominator has a $$3$$ and $$4$$ and the numerator has $$25$$ which mean a $$3-4-5$$ triangle might be related to this integral but I'm not to sure.

Any help is appreciated :)

• Hint: $3\sin(x)+4 \cos(x)=5 \sin(x+\arctan(4/3))$ – tired Dec 20 '16 at 12:09
• Furthermore $\int dx \sin(x)^{-2}=\text{cotan(x)}$ – tired Dec 20 '16 at 12:12

Hint. One may write $$\int \frac{25}{(3\cos(x)+4\sin(x))^2} dx=\int \frac{25}{(3+4\tan(x))^2} \frac{dx}{\cos^2 x}=\int \frac{25}{(3+4u)^2} \:du$$ with $u=\tan x$.
Let $\varphi$ such that $\;\begin{cases}\cos\varphi=\frac35,\\\sin\varphi=\frac45.\end{cases}$, e.g. $\;\varphi=\arctan\bigl(\frac43\bigr)$.
The integral can be rewritten as $$\int\frac{\mathrm dx}{\cos^2(x-\varphi)}=\tan(x-\varphi).$$