If x^2 1-x ~d x= p 1-x (3 x^2+4 x+8 )+ c where c is a constant of integration, then the value of p is

Step-by-step solution

aligned & Let I = x^2 1-x ~d x & Put 1-x= t & d x=- dt & I = (1- t )^2 t (- dt ) & =- 1-2 t + t ^2 t ^ 1 2 dt & =- (t^ - 1 2 -2 t^ 1 2 +t^ 3 2 ) d t & =- (2 t^ 1 2 - 4 3 t^ 3 2 + 2 5 t^ 5 2 )+c & =t^ 1 2 ( -30+20 t-6 t^2 15 )+c & aligned aligned & = 1-x ( -30+20(1-x)-6(1-x)^2 15 )+ c & = 1-x ( -6 x^2-8 x-16 15 )+ c & =- 2 15 1-x (3 x^2+4 x+8 )+ c p & = -2 15 aligned