why substitution works in integration ?

 

The first thing that we must know is that integration and differentiation are reverse process of each other. So, every rule in integration is somewhere related to the rules in differentiation.

 

The reason, why substitution works is due to the chain rule in differentiation. The following will build an intuition on how chain rule connects to the substitution. For indefinite integration. (note:- using substitution to evaluate a definite integral requires a change to the limits of integration)

 

Suppose we have two continuous functions f(x) and g(x),the derivative of f[g(x)] is as follows

 

`frac{\text{d}f[g(x)]}{\text{d}x}= f'[g(x)].g'(x)`

If we integrate both sides with respect to x we get -

 

f[g(x)] + c =  `\int_{}^{} f'[g(x)].g'(x)`

(RHS)                     (LHS)

 

Let take a look at (LHS), if we substitute u= g(x) then `\frac{\text{d}u}{\text{d}x}` = g’(x) and du= g’(x)dx , so substituting this in (LHS) we get -

 

`\int_{}^{}f'(u)du=f(u)+c `  

 

And putting back the value u= g(x) we get back the (RHS)