If u and v are functions of x , then 1 v^3 (uv du dx - u^2 dv dx )dx =

Step-by-step solution

The given integral is I = 1 v^3 (uv du dx - u^2 dv dx )dx Rearranging the terms in the integrand, we get: I = u v ( v du dx - u dv dx v^2 )dx We know that by the quotient rule of differentiation, d dx ( u v ) = v du dx - u dv dx v^2 Substituting this into the integral: I = u v d dx ( u v ) dx Let u v = t , then d dx ( u v ) dx = dt I = t dt I = t^2 2 + c Substituting back t = u v : I = u^2 2v^2 + c Answer: u^2 2v^2 + c