Question

Is there an inverse chain rule for integration?

Answer 1

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`