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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:

$\int \sin( 5 x + 4 ) \ dx$

A first guess of the antiderivative might be:

$-\cos( 5 x + 4 ),$

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

$\frac{ d }{ dx } \left( -\cos( 5 x + 4 ) \right) \; = \; 5\sin(5 x + 4)$

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

$\int \sin( 5 x + 4 ) \ dx \; = \; -\frac{1}{5} \cdot \cos( 5 x + 4 ) + C$

This method can be used to find:

$\int f( g(x) ) \; dx$

and g(x) is a linear function.