If x^5 e ^ -4 x^3 ~d x= 1 48 e ^ -4 x^3 f (x)+ c , where c is a constant of integration, then f (x) is given by

Step-by-step solution

aligned & Let I = x^5 e ^ -4 x^3 ~d x & aligned Put x^3= t 3 x^2 ~d x & = dt I = 1 3 te ^ -4 t dt & = 1 3 ( t e ^ -4 t -4 - 1 e ^ -4 t -4 dt ) & = 1 3 ( - te ^ -4 t 4 + 1 4 e ^ -4 t -4 )+ c & = 1 48 e ^ -4 t (-4 t -1)+ c & = 1 48 e ^ -4 x^3 (-4 x^3-1 )+ c aligned aligned f (x)=-4 x^3-1