View source for Talk:Integration by substitution

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== integrability of phi ==

Is it necessary to suppose integrability of (phi)'?
Isn't it guaranteed by its assumption of continuity?

== most general set of conditions ==

I'm thinking that in order to be a definitive source on the subject, there should be a theorem
which states, in <math>\mathbf{R}^1</math>, the most general set of circumstances for F under which the change of
variables works in the Reimann-Stiltjes sense

<math> \int_a^b \psi(F(t)) dF(t) = \int_{F(a)}^{F(b)} \psi(\xi) d\xi </math>

that is to say that in order for the left hand side to exist as a Reimann-Stiltjes integral, the
<math>\psi(F(t))</math> must be integrable on (a,b) and the function, F, must be of bounded variation.
Nothing about differentiability, and far less, nothing even about absolute continuity.


According to an exercise in Royden, the above is true for all F continuous and increasing
(no need for absolute continuity and of course the requirement that <math>\psi(F(t))</math> 
is integrable). Thus the above change of variables works when F is the Cantor function. 

The Hewitt & Stromberg (1965, Theorem 20.3) citation attempts to be a general reference--its
stated for functions on Polish Spaces, which is fine but more general than required. Something
gives me the suspicion that this synopsis is mis-stated (c.f. the requirement that a function
must be continuous _and_ absolutely continuous is redundant) also absolute continuity is not strictly
required. [[User:Izmirlig|Izmirlig]] ([[User talk:Izmirlig|talk]])izmirlig@mail.nih.gov

::Someone signed incorrectly. [[User:Nerd271|Nerd271]] ([[User talk:Nerd271|talk]]) 19:14, 4 April 2020 (UTC)

== A section for discussing the geometry of trigonometric substitutions? ==

So consider the arctangent integral, 
:<math> 
\int \frac{dx}{1+x^2}
</math>
The "standard" substitution is <math>u(x) = \tan(x)</math>, which then simplifies the integral to integration of a constant function via the derivative, <math>\sec^2(x)</math>.

We see is is useful by direct computation, but one could also look at a right triangle, with angle <math>x</math> and side lengths <math>1,u(x),\sqrt{1+u(x)^2}</math>.

This corresponds to the form of the integrand itself. From here, it is easy to see that <math>u(x) = \tan(x) \text{ and that } u'(x) = \sec^2(x)</math>, which leads to our desired solution.


I think this perspective is important in teaching the method of substitution not as a rote algebraic trick, but one motivated by geometric intuition. [[User:Legatostaccato|Legatostaccato]] ([[User talk:Legatostaccato|talk]]) 06:40, 11 February 2020 (UTC)

:We do have a page for trigonometric substitutions. [[User:Nerd271|Nerd271]] ([[User talk:Nerd271|talk]]) 19:14, 4 April 2020 (UTC)

== Non-injective case ==

I think we should mention the possibility to relax the injectivity condition, see [https://www.planetmath.org/changeofvariablesinintegralonmathbbrn]Theorem 2 or Theorem 5.8.30 in ''Measure Theory - Volume I'' (Bogachev).

The question naturally arises when comparing with the univariate case.

I don't feel comfortable enough to write it however. [[User:Trashyyy|Trashyyy]] ([[User talk:Trashyyy|talk]]) 12:26, 10 November 2024 (UTC)