Tuesday, 22 January 2019

calculus - Need some advice to solve this integral $intfrac{sin^2x}{1+sin^2x}mathrm dx$




I'm trying to use this subtitution $t=\tan(x/2)$. But I don´t get anywhere. I've tried $t=\tan(x)$ too. Appreciate your help.



$$\int\dfrac{\sin^2x}{1+\sin^2x}\mathrm dx$$


Answer



You can use the substitution $x=\arctan(t/2)$ and you will need the identity




$$ \sin( \arctan(t/2) ) = \frac{t}{\sqrt{t^2+1}} $$





to reach the form




$$ I= \int \frac{t^2}{(t^2+2)(t^2+4)}dt. $$




I think you can finish it now!


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