Find the indefinite integral $\int \frac{25}{(3\cos(x)+4\sin(x))^2} dx$
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 $$\int\frac{\mathrm dx}{\cos^2(x-\varphi)}=\tan(x-\varphi).$$