Question

How is integration by substitution related to the chain rule?

Answer 1

Now we know that integration by substitution is nothing but substituting a function by a variable and the integrating for simplicity Hence instead of g(f(x))f’(x) we can write it as g(u)du where u = f(x). Similarly we know that the chain rule of differentiation says that $\dfrac{d\left( f\left( g\left( x \right) \right) \right)}{dx}=f'\left( g\left( x \right) \right)g'\left( x \right)$ Hence we will compare the two and understand the relation between the them.

Complete step-by-step answer:
Now first let us understand the concept of integration by substitution and chain rule.
Now chain rule is a rule in differentiation which helps us to differentiate composite function.
According to chain rule we have $\dfrac{d\left( f\left( g\left( x \right) \right) \right)}{dx}=f'\left( g\left( x \right) \right)g'\left( x \right)$
Now let us understand the concept of integration by substitution.
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 then substitute the value of u.
For example consider the integration $\int{{{x}^{2}}2xdx}$
To do this integration we will substitute ${{x}^{2}}=u$ then differentiating we get $2xdx=du$ .
Hence on substitution we get the integral as $\int{udu}$ and hence we can easily integrate and re substitute the value of u.
Now integration is nothing but the reverse of differentiation.
Similarly integration by substitution is nothing but reverse of chain rule.
In chain rule we derive $f\left( g\left( x \right) \right)$ as $f'\left( g\left( x \right) \right)g'\left( x \right)$ while in integration by substitution we take the expression of the form $f'\left( g\left( x \right) \right)g'\left( x \right)$ and then find its antiderivative as $f\left( g\left( x \right) \right)$ .
Hence integration by substitution is nothing but reverse of chain rule.

Note: Note that when we substitute f(x) = u we also convert the differential element dx to du by differentiation the equation f(x) = u. Hence remember not to just replace dx by du but find the proper substitution for it. Also note that if there are two or more functions in chain rule we follow the same method. For example differentiation of ${{e}^{\left( \sin \left( {{x}^{2}} \right) \right)}}$ is ${{e}^{\left( \sin \left( {{x}^{2}} \right) \right)}}.\cos \left( {{x}^{2}} \right).2x$ .