If f(x) , dx = g(x) then x^3 f(x^2) , dx is equal to

Step-by-step solution

Let I = x^3 f(x^2) , dx . Substitute x^2 = t , which gives 2x , dx = dt or x , dx = 1 2 dt . I = x^2 f(x^2) x , dx = 1 2 t f(t) , dt Using integration by parts, taking t as the first function and f(t) as the second function: I = 1 2 [ t f(t) , dt - ( d dt (t) f(t) , dt ) dt ] Since f(x) , dx = g(x) , we have f(t) , dt = g(t) . I = 1 2 [ t g(t) - g(t) , dt ] Substituting back t = x^2 and dt = d(x^2) : I = 1 2 [ x^2 g(x^2) - g(x^2) , d(x^2) ]