Integration Product Rule

 Integration by parts is defined as a rule that converts the integration of the product the functions into other easier integrals. The integration by parts is also called as the anti product rule, because the integration by parts appears from the product rule of differentials.

Let us consider the two functions which are given by u = f(x) and v=g(x), then the product rule for the integration is given by

                                         ∫u(dv/dx) dx=uv-∫v(du/dx) dx

 

 

Product rule for integration:

 

Let us consider two continuous differentiable functions such as f(x) and g(x), then according to the product rule the two functions are given by,

                     (f(x)*g(x))′ = f(x)g′(x) + f′(x)g(x)

If we take integration on both sides then we get,

                  f(x)g(x) = ∫ f(x)g′(x) dx+∫ f′(x)g(x) dx.

If we arrange the terms we get,

                 ∫ f(x)g′(x) dx = f(x)g(x) - ∫ f′(x)g(x) dx

Let us consider the closed interval a and b for the above functions then,

                 b                                  b  b
                 ∫ f(x)g′(x) dx = [f(x)g(x)] - ∫ f′(x)g(x) dx
                a                                   a  a

If we explain it with the common terms then we get,

                          b
              [f(x)g(x)]  = f(b)g(b) – f(a)g(a)
                          a

                                                b
              f(b)g(b) – f(a)g(a) = ∫ d/dx f(x)g(x) dx
                                                a

                                   b                                  b  b
                               = ∫ f′(x)g (x) dx = [f(x)g(x)] - ∫ f′(x)g(x) dx
                                  a                                   a  a

The product rule for the integration can also be stated in the form given by

                                ∫ f (x)g′(x) dx = f(x)g(x) - ∫ f′(x)g(x) dx

We already know that f(x) = u and g(x) = v and du = f′(x) and dv = g′(x)

                                  ∫u(dv/dx) dx=uv-∫v(du/dx) dx

 

Rule using the summation by parts:

 

 

Let us consider an alternative notation for the integration that has been given by f and g

                      b                         b   b  b
                      ∫  fg dx = [f ∫ g dx] - ∫ - ∫ ( ∫g dx)df
                     a                          a  a   a

This is said to be true when the function f is continuously differentiable and g is continuous. The difficult form of the product rule for the integration is given by

                  ∫ uv dw = uvw - ∫ uw dv - ∫ vw du