Question

Is there an inverse chain rule for integration?

Answer 1

Integration is the opposite process of the differentiation. If differentiation chain rule allows the differentiation of functions that are composite, we can denote chain rule by $f.g$ where $f$ and $g$ are the two functions. Here we will see which rule of integration can be called an inverse chain rule.

Complete answer:
Integration is a method of adding or summing up the parts to find the whole. It is a reverse process of differentiation, where we reduce the functions in smaller parts. Differentiation is the process of finding the derivative and integration is the process of finding the antiderivative of a function. So, these processes are inverse of each other. There are various methods to integrate the function such as integration by substitution, integration by parts, integration by partial fraction and integration using trigonometric identities.
Now, first let us understand the concept of chain rule and then we will see which methods of integration can be called as inverse chain rule.
Chain rule is a rule in differentiation which allows us to differentiate composite functions.
According to chain rule we have
$\dfrac{{d(f(g(x)))}}{{dx}} = f'(g(x))g'(x)$
Now, in integration the method of integration can be called the inverse chain rule as integration by substitution is a method of substitution in which we substitute the function $f(x) = u$ and then integrate the function with respect to $du$ and then substitute the value of $u$.
For example- consider the integration $\int {{x^2}} 2xdx$
For integration we will substitute ${x^2} = u$.
Then differentiating the function with respect to $x$ we get $2xdx = du$
Hence, on substation we get the integral as $\int {udu}$ and now we can easily integrate and re substitute the value of $u$.
As integration is nothing but the reverse of differentiation. So, integration by substitution is nothing but the inverse of chain rule.
In chain rule we find the derivative of function as $f'(g(x))g'(x)$ while in integration by substitution we take the expression of the form $f'(g(x))g'(x)$ and then find its antiderivative as $f(g(x))$.
Hence, integration by substitution is the reverse of chain rule.

Note: If we are substituting $f(x) = u$ we must have to convert the differential element $dx$ to $du$ by differentiating the equation $f(x) = u$. In integration by substitution method we have to find the proper substitution for it if there is not a proper differentiation, use another method for finding the integration of the function and don’t remember to differentiate the function which is to be substituted.