x ^3 1+ x ^2 dx = a (1+ x ^2 )^ 3 2 + b 1+ x ^2 + c , (where c is constant of integration) then find the value of a+b

Step-by-step solution

Let I= x^3 1+x^2 d x= x^2 x 1+x^2 d x Put 1+ x ^2 = t 1+ x ^2= t ^2 2 xdx =2 t d t I = ( t ^2-1 ) t dt t = ( t ^2-1 ) dt = t ^3 3 - t + c = (1+ x ^2 )^ 3 2 3 - 1- x ^2 + c Comparing with given data, a = 1 3 , ~b =-1 a + b = -2 3