Evaluate $ \int \frac{\tan(x)}{2+\sin(x)}dx $

Question

How do you evalute this integral? $$ \int \frac{\tan(x)}{2+\sin(x)}dx $$

Answer 1

HINT:

$$\frac{\sin x}{\cos x(2+\sin x)}=\frac{\sin x+2-2}{\cos x(2+\sin x)}$$

$$=\sec x-2\cdot\frac{\cos x}{(1-\sin^2x)(2+\sin x)}$$

For the second integral, set $\sin x=u$

$$\int\frac{\cos x}{(1-\sin^2x)(2+\sin x)}dx=\int\frac1{(1-u)(1+u)(2+u)}du$$

Method$\#1:$ Now use Partial Fraction Decomposition

Method$\#2:$ $\dfrac1{(1-u)(1+u)(2+u)}=\dfrac{2+u-(1+u)}{(1-u)(1+u)(2+u)}$

$=\dfrac1{(1-u)(1+u)}-\dfrac1{(1-u)(2+u)}$

Again $\dfrac1{(1-u)(1+u)}=\dfrac12\dfrac{1-u+1+u}{(1-u)(1+u)}=$

and $\dfrac1{(1-u)(2+u)}=?$