Skip to content

Files

Latest commit

39d2c2b · Nov 12, 2024

History

History
76 lines (54 loc) · 1.82 KB

change_of_variables_int.md

File metadata and controls

76 lines (54 loc) · 1.82 KB
tags
math

Change of variables in integrals

Indefinite integrals

Imagine an integral with an expression f ( x ) that you want to substitute for y :

G ( x ) = g ( f ( x ) ) d x

What this tells us that if we derivate G ( x ) we would get g ( f ( x ) ) . We are free to substitute x for anything else (at both sides of the equation), but we need to be vary of the chain rule which is more visible if we rewrite the above as differentiation:

G ( x ) = g ( f ( x ) )

If we want to get rid of f ( x ) by substituing it for y := f ( x ) we essentially substitute x := f 1 ( y ) . If we do that on the LHS, we need to multiply RHS by ( f 1 ) ( y ) according to the chain rule, since RHS is differentiation of LHS:

G ( f 1 ( y ) ) = g ( f ( f 1 ( y ) ) ) f 1 ( y ) y = g ( y ) 1 f ( x )

Where the last equation is just rule of differentiating of functions, derived from derivating x = f ( f 1 ( x ) ) on both sides.

Finally, the last equation in integral form:

G ( f 1 ( y ) ) = g ( y ) 1 f ( x ) d y

To summarize:

  1. identify the function which we want to get rid of ($f(x)$)
  2. realize what we are substituing x for ( x for $f^{-1}(y)$)
  3. realize we are substituing in the differentiating RHS and we need to follow the chain rule (multiply by ( f 1 ) ( y ) or divide by $f^\prime(x)$)

Example

Let's say we substitute y := 2 x 3 + 1 in:

( 2 x 3 + 1 ) 7 x 2 d x

Now:

2 x 3 + 1 x = 6 x 2

So we need to divide the expression by 6x:

( 2 x 3 + 1 ) 7 x 2 d x = ( y ) 7 x 2 6 x 2 d y = 1 6 y 7 d y

Definite integral

If we have definite integral, say going from a to b and we substitute y := f ( x ) , then we proceed as with indefinite integral, only changing a to f ( a ) and b to f ( b ) .