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:
- <math>
\int \sin( 5 x + 4 ) \ dx
</math>
A first guess of the antiderivative might be:
- <math>
-\cos( 5 x + 4 ),
</math>
treating (5x+4) as if it were an x. Differentiating back with the chain rule gives:
- <math>
\frac{ d }{ dx } \left( -\cos( 5 x + 4 ) \right) \; = \; 5\sin(5 x + 4)
</math>
Hence, the initial guess was off by a factor of 5. Dividing by 5 gives:
- <math>
\int \sin( 5 x + 4 ) \ dx
\; = \; -\frac{1}{5} \cdot \cos( 5 x + 4 ) + C
</math>
This method can be used to find:
- <math>
\int f( g(x) ) \; dx
</math>
and g(x) is a linear function.