The function y=f(x) is the solution of the differential equation dydx+xyx2−1=x4+2x1−x2 in x∈(−1,1) satisfying f(0)=0 the ∫−323/2 f(x)dx is

I.F.= e∫xx2−1dx=e12log⁡x2−1∣=1−x2 Solution of D.E is y1−x2=∫x4+2x1−x2⋅1−x2dy=x55+x2+cf(0)=0⇒c=0y1−x2=x55+x2∫−3232 y dx=∫−3232 x21−x2dx     If f(-x)=f(x) then ∫-aaf(x) dx=2∫0af(x) dx=2∫032 x21−x2dx, put x=sin⁡θ  ⇒dx=cosθ dθ =2∫0π/3 sin2⁡θ dθ=2∫1-cos2θ2dθ=θ-sin2θ20π3=π3-34