Is there an inverse chain rule for integration?

1 Answer
Oct 29, 2015

Answer:

Integration by substitution is the inverse of differentiation using the chain rule.

Explanation:

#intf(g(x))g'(x)dx#

Let #u = g(x)#. This make #du=g'(x) dx# and the integral becomes

#intf(u)du#

A different way to see this is to do n integration by substitution and then check the answer by differentiation.

Example

#int 3x^2 sqrt(x^3+7) dx#

Let #u = x^3+7#. This makes #du = 3x^2 dx# and the integral becomes:

#int 3x^2 sqrt(x^3+7) dx=int (underbrace(x^3+7)_u)^(1/2) underbrace(3x^2dx)_(du) =int u^(1/2)du#

# = 2/3u^(3/2)+C = 2/3(x^3+7)^(3/2)+C#

Now, if we check our answer by differentiating, we will use the chain rule.

We could also write the integration using #g(x)# rather than #u#.

#int 3x^2 sqrt(x^3+7) dx = int (x^3+7)^(1/2) 3x^2dx#

With #g(x) = x^3+7#, we have #g'(x) = 3x^2#, so the integral is

# = int (g(x))^(1/2) g'(x)dx = 2/3(g(x))^(3/2)+C#

#= 2/3(x^3+7)^(3/2)+C#