If ^4 x dx =a ^3 x+ b x+ c x+ k (where k is the constant of integration) then the value of a-b+c=

Step-by-step solution

Consider the integral I = ^4 x dx . Rewriting using the identity ^2 x = ^2 x - 1 : I = ^2 x ( ^2 x - 1) dx = ^2 x ^2 x dx - ^2 x dx For the first term, substitute u = x , du = ^2 x dx : u^2 du = u^3 3 = ^3 x 3 For the second term, apply the identity again: ^2 x dx = ( ^2 x - 1) dx = x - x The complete integral becomes: I = ^3 x 3 - x + x + k Matching with the given form a ^3 x + b x + c x + k yields coefficients a = 1 3 , b = -1 , c = 1 . Therefore: a - b + c = 1 3 - (-1) + 1 = 7 3