Question

Evaluate $\int \frac{5\cos x+6}{2\cos x+\sin x+3}dx$

Answer 1

$\begin{array}{c}Wehave\\ \int \frac{5\cos x+6}{2\cos x+\sin x+3}dx\\ Here,\\ \left(5\cos x+6\right)=\lambda \left(2\cos x+\sin x+3\right)+\mu \left(-2\sin x+\cos x+u\right)\\ =\left(2\lambda +u\right)\cos x+\left(\lambda -2\mu \right)\sin x+\left(3\lambda +u\right)\\ \therefore 2\lambda +u=5,\lambda -2\mu =0,3\lambda +u=6\\ so,\\ \lambda =2\\ \mu =1\\ u=0\\ I=\int \frac{2\left(2\cos x+\sin x+3\right)+\left(\cos x-2\sin x\right)}{\left(2\cos x+\sin x+3\right)}dx\\ =2\int dx+\int \frac{\left(\cos x-2\sin x\right)}{\left(2\cos x+\sin x+3\right)}dx\\ I=2x+\log \left(2\cos x+\sin x+3\right)+C\end{array}$