Answer :
[tex]I= \int\limits^{}_{} {\frac{Sin\ 2x}{a^2+b^2Sin^2x}} \, dx \\\\=\int\limits^{}_{} {\frac{2Sin\ x}{a^2+b^2Sin^2x}} \,\ (cos\ x dx) \\\\Let\ t=Sin\ x,\ \ \ dt=Cosx\ dx\\\\I=\frac{1}{b^2} \int\limits^{}_{} {\frac{2b^2\ t}{a^2+b^2t^2}} \, dt \\\\=\frac{1}{b^2}Ln(a^2+b^2t^2)\\\\=\frac{1}{b^2}Ln(a^2+b^2Sin^2x)[/tex]