Question

Evaluate: $\int \frac{dx}{\cos x+\sin 2x}$

Answer 1

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

$\Rightarrow \int \frac{dx}{\cos x(1+2\sin x)}=\frac{\cos xdx}{{\cos }^{2}x(1+2\sin x)}$

$\Rightarrow \frac{\cos xdx}{(1-{\sin }^{2}x)(1+2\sin x)}$

$\sin x=t$

$\cos xdx=dt$

$\Rightarrow \frac{dt}{(1-t^2)(1+2t)}=\frac{dt}{(1+t)(1-t)(1+2t)}$

Breaking into partial fractions

$\Rightarrow [-\frac{dt}{2(1+t)}+\frac{4dt}{3(1+2t)}+\frac{dt}{6(1-t)}]$

$=\frac{-1}{2}\ln |1+t|+\frac{2}{3}\ln |1+2t|-\frac{1}{6}\ln |1-t|$

$=\frac{-1}{2}\ln |1+\sin x|+\frac{2}{3}\ln |1+2\sin x|-\frac{1}{6}\ln |1-\sin x|$