If 2x ^4 x + ^4 x , dx = f(x) + c where c is constant of integration, then the value of (f(x)) is

Step-by-step solution

The given integral is 2x ^4 x + ^4 x , dx . Dividing the numerator and the denominator by ^4 x , we get 2 x x ^4 x ^4 x + ^4 x ^4 x , dx = 2 x ^2 x ^4 x + 1 , dx Substituting ^2 x = t , we get 2 x ^2 x , dx = dt . The integral becomes dt t^2 + 1 = ⁻¹(t) + c Substituting back t = ^2 x , we get f(x) + c = ⁻¹( ^2 x) + c Comparing both sides, f(x) = ⁻¹( ^2 x) . Therefore, (f(x)) = ( ⁻¹( ^2 x)) = ^2 x . Answer: ^2 x