Edited by Valentin Fadeev, Sunday, 16 Jan 2011, 23:18
Some integrals yield only one type of substitution that really brings them into a convenient form. Any other method would make them more complicated. However, in some cases totally different methods can be applied with equal effect. In case of definite integrals it is of course not necessary to come back to original variable which makes things even easier. Here is one example
The most natural way is to apply a trigonometric substitute. We will not consider this method here. Instead an algebraic trick can be employed:
Alternatively we can use integration by parts:
Or apply an even more exotic treatment:
let
let
MathJax failure: TeX parse error: Extra open brace or missing close brace
On variable changes in definite integrals
Some integrals yield only one type of substitution that really brings them into a convenient form. Any other method would make them more complicated. However, in some cases totally different methods can be applied with equal effect. In case of definite integrals it is of course not necessary to come back to original variable which makes things even easier. Here is one example
The most natural way is to apply a trigonometric substitute. We will not consider this method here. Instead an algebraic trick can be employed:
Alternatively we can use integration by parts:
Or apply an even more exotic treatment:
let
let
MathJax failure: TeX parse error: Extra open brace or missing close brace
For
and
,
the same result is obtained