Inverse chain rule method

In calculus, the inverse chain rule is a method of integrating a function which relies on guessing the integral of that function, and then differentiating back using the chain rule. The method is a special case of integration by substitution.

For example, suppose one wants to find the indefinite integral:

A first guess of the antiderivative might be:

treating (5x+4) as if it were an x. Differentiating back with the chain rule gives:

Hence, the initial guess was off by a factor of 5. Dividing by 5 gives:

This method can be used to find:

and g(x) is a linear function.