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Imagine an integral with an expression that you want to substitute for
:
What this tells us that if we derivate we would get . We are
free to substitute 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:
If we want to get rid of by substituing it for we essentially
substitute . If we do that on the LHS, we need to multiply RHS
by according to the chain rule, since RHS is
differentiation of LHS:
Where the last equation is just rule of differentiating of functions, derived
from derivating on both sides.
Finally, the last equation in integral form:
To summarize:
identify the function which we want to get rid of ($f(x)$)
realize what we are substituing for ( for $f^{-1}(y)$)
realize we are substituing in the differentiating RHS and we need to follow
the chain rule (multiply by or divide by $f^\prime(x)$)
Example
Let's say we substitute in:
Now:
So we need to divide the expression by 6x:
Definite integral
If we have definite integral, say going from to and we substitute , then we proceed as with indefinite integral, only changing to
and to .