Making trigonometric substitutions rigorous

I’ve been tutoring some basic calculus, and it made me think about something pretty basic.

Let me explain the problem by example:

Say we are given the integral x21x2 dx. It is then customary to write x=cos(α),dx=sin(α) dα. So:
cos2(α)sin2(α))(sin(α)) dα=cos2(α)sin(α)sin(α) dα=cos2(α) dα=1+cos(2α)2 dα=α2sin(2α)4=α2sin(α)cos(α)2.

We know plug in α=arccos(x). It is left to know what sin(arccos(x)) is.

For this, we now we pretend that 0απ2 and draw a triangle which shows via the Pythagorean theorem that sin(arccos(x))=1x2.

Similarly there are the common substitutions x=tan(α) and x=sec(α).

This technique seems less than rigorous. Here are some of my issues with it:

  1. At first it seemed to me like this is u-substitution in reverse (think that x plays the role of u and α plays the role of x). But in u-substitution, if the substitution is u=g(x) then we eventually plug in g(x) instead of u. But here the roles are reversed. So it seems that this really is ordinary u-substitution. But then that means that the substitution is α=arccos(x) — however there are many ways to choose inverses to cos and arccos(x) is only one of them. Are we really choosing just that one?

  2. I feel uncomfortable with sin2(x)=sin(x). What if sin(x) is negative? What’s going on here? Is the issue that we are just doing it on a segment where sin(x) is positive, do the whole thing, and then after we get an answer we do some argument using analytic continuation?
    Or is perhaps the issue that we are looking at 1<x<1 (because that's where 1x2 is defined), and so arccos(x) would go from 0 to π where sin(x) is positive? Would this weird argument also work for other trigonometric substitutions?

  3. That part at the end where we figure out what the trigonometric function of an inverse trigonometric function is (in my example sin(arccos(x)) is fishy to me. To figure it out I draw a triangle, which seems to assume arccos(x) is between 0 and π2. What's going on there.

So, while I'm very familiar with the method, tutoring it made me realize I'm not sure what's really behind it. Could it really be something as complicated as analytic continuation behind it? Is there a uniform way of explaining this method with respect to the substitutions x=cos(α),x=tan(α) and x=sec(α)?

Answer

  1. Yes, in a sense it's like a u-substitution "in reverse". However, there is really no problem here as long as you are careful (this will also take care of your issue in 2).

    You can think of the integral you are seeing as what you would have gotten if you had started with the integral
    sinαcosαdα1cos2α,
    and had tried the substitution u=cosα; and the trig substitution merely gets you back to this after a substitution that didn't work out so well.

    In your example, for the integral to be defined you must have x(1,1). This means that each value of x can be written uniquely as x=cos(α) for some α(0,π). Now, because we want the substitution to be reversible, we have to specify this. But this also ensures that the "correct" inverse is arccos(x).

  2. Now, indeed, sin2α=|sinα|, not sinα. But, remember that we were restricting α to lie in (0,π). In that interval, |sinα|=sinα, so we can indeed drop the absolute value bars. And yes, this argument also works for any trigonometric substitution: if you are going from a2x2, you need x to lie on [a,a] or (a,a), so the substitution x=asinα with π/2απ/2 ensures the cosine is positive; the substitution x=acosα with α[0,π] ensures sine is positive.

    When using the substitution x=atanθ, you likewise restrict to the "main" branch of the tangent, on π/2<θ<π/2, where the inverse is arctan(x), and the secant is positive (so that 1+tan2θ=secθ).

    The tricky one is x=asecθ; you want to restrict to [0,π/2)(π/2,π]. When there are no radicals involved, you probably want to avoid this identity anyway; when there are radicals, like x2a2, you need to take into account that the domain consists of two disjoint intervals, (,a] and [a,); you use 0θ<π/2 on the latter, and π/2<θπ for the former. Which one you are in determines whether tan(θ) is positive or negative. Added. Sometimes you want to pick different intervals to ensure the signs work out right, though.

  3. Drawing the triangle is a simple mnemonic, but it can be justified purely algebraically.
    You want to find sin(arccos(α)). Well,
    1=sin2(arccos(α))+cos2(arccos(α))=sin2(arccos(α))+α2,
    hence sin2(arccos(α))=1α2. Remembering that arccos(α) is always on [0,π], it follows that sin(arccos(α))=1α2. Similar manipulations of the standard trigonometric identities work for other trig substitutions.

Attribution
Source : Link , Question Author : Amy , Answer Author : Arturo Magidin

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