Inverse chain rule method

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In calculus, the inverse chain rule is a method of integrating a function which relies on guessing the integral of that function, and then checking that the integral is correct by differentiating back using the chain rule. If it is not correct, compensation is made with constants. 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.

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Category: Integral calculus

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