There is a "Chain – Rule" in calculus that allows us to differentiate a composite function in the following way (Sorry for mixing up the two standard derivative notations.) :
$$\frac {d}{dx} f(g(x)) =g'(x) . f'(g(x))$$
Does there exist a similar "(Reverse) Chain Rule" for Integration ?
$$\int f(g(x)) dx = ?$$
Wolfram Alpha says : "no result found in terms of standard mathematical functions" and gives a very horrible "Series expansion of the integral at x=0".
So I expect a negative response, but then how can expressions be integrated : $\sin(nx),\cos(nx) , etc.$
It is obvious that some other techniques like u-substitution, trigo substitution , by parts, etc. have to be applied, but does there exist "something" in this universe which provides a general formula for integrating composite functions ?
Best Answer
Yes, you are partly correct. Sometimes there are composite functions like you have mentioned, which can be integrated using traditional methods to obtain elementary functions as their antiderivative. But then again, you have certain composite functions like $e^{x^2}$, $\ln(x^2+x)$ which do not have proper antiderivatives expressible in terms of elementary functions. So, in order to determine which function, or rather composite function, can be integrated indefinitely to obtain a result in terms of elementary functions, we can use the Risch Algorithm which tests whether a certain function can be integrated indefinitely and also gives the procedure of obtaining that integral.
Hope this helps you.