Question

$\int \frac{\sin 2x}{(a{\sin }^{2}x+b{\cos }^2{)}^2}dx$

Answer 1

$\int \frac{sin2x}{(asi{n}^{2}x+bco{s}^{2}x{)}^2}dx$

$si{n}^{2}x3t$

$si{n}^{2}xd{x}^{3}dt$

$\therefore I=\int \frac{1}{(at+b(1-t){)}^2}dt$

$=\int \frac{1}{(at+b-bt{)}^2}dt$

$=\int \frac{1}{\left(\right(a-b)t+b{)}^2}dt$

$=\frac{1}{(a-b)}-\frac{1}{(a-b)t+b}+c$

$=-\frac{1}{(a-b)\left(\right(a-b)si{n}^{2}x+b)}+c$