Talk:Integral of the secant function
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Equivalent forms
[edit]Hi!
Can someone show how the first two forms are equivalent? It'd be much appreciated.
Thanks, Wisapi (talk) 17:58, 27 July 2011 (UTC)
(secθ+tanθ)^2=((1+sinθ)/cosθ)^2 =(1+sinθ)^2/(1-(sinθ)^2) =(1+sinθ)^2/(1+sinθ)(1-sinθ) =(1+sinθ)/(1-sinθ)
Hence ln((secθ+tanθ)^2)=2ln(secθ+tanθ)=ln((1+sinθ)/(1-sinθ)) ∴ ln(secθ+tanθ) = 1/2 ln((1+sinθ)/(1-sinθ)) QED
Hope this helps — Preceding unsigned comment added by 131.111.185.74 (talk) 15:38, 21 October 2011 (UTC)
Further developments
[edit]I have made some additions. Please check for typos.
As it stands the article uses the phrase "solved by" in relation to both Gregory and Barrow. It should be made clear that these were probably independent solutions. It is also probable that Halley had a solution too. It would be nice to have (accessible?) references to these solutions: presumably their methods differed. Is the solution presented here that of Barrow or Gregory?
The three forms of the solution given do not exhaust all possibilities. There are two more on the Mercator projection page. (Alternatively see my draft revision of that page here. These forms involving hyperbolic functions are much used in current work on projections. They should be added to this page. Peter Mercator (talk) 14:54, 30 May 2012 (UTC)
Now added. Peter Mercator (talk) 18:07, 6 January 2015 (UTC)
Link needed
[edit]A equation linking theta and u is needed. — Preceding unsigned comment added by 86.176.146.34 (talk) 16:41, 3 January 2015 (UTC)
Partial fractions
[edit]@Sraianu: Partial fraction decomposition is Pre-calculus, even Algebra II, material. I'm not sure you can really call your algebraic trick an alternative to partial fraction decomposition. It appears somewhat unusual rather than systematic. Moreover, what you are actually doing is to break down a complicated fraction into simpler ones. What is that if not partial fraction decomposition, albeit done in a non-standard way? Nerd271 (talk) 17:08, 18 July 2020 (UTC)
- Regardless, it's WP:OR, and I've re-removed it as such. In any case, it's such a minor difference in approach as to not bother with. On a side note, please also see MOS:TONE – phrasing like "We note that..." isn't ideal. –Deacon Vorbis (carbon • videos) 17:26, 18 July 2020 (UTC)
- Partial fractions, or the partial fractions method, has a precise meaning, it is actually a theorem in abstract algebra and the proof uses arithmetic in the ring of polynomials, you can see a proof of it in my book on my webpage (just google my name, then search for partial fractions in the pdf, or use the index, clicking on the page number in the index will take you there). In North America partial fractions are usually taught in Calculus II, Integral Calculus, right after trigonometric integrals and substitutions, see for example Stewart's book (any edition or version), or Thomas Calculus (or Finney and Thomas, or any combination, any edition). I agree that "We note that" is not ideal, you can change that to something better if you want, but I would like my edit to remain, for the teaching implications that I tried to describe before, I think that even the fact that you call the trick of adding and subtracting x (well, u rather) unusual is an argument for leaving my edit in. Can you show this dispute to other editors (the more of them teaching calculus the better) so that they can vote on this? Thanks, Sraianu (talk) 17:53, 18 July 2020 (UTC)
"Can you show this dispute to other editors (the more of them teaching calculus the better) so that they can vote on this?"
. Please also see WP:NOTAVOTE. If you want to get a bit more outside input, a first step would probably be to (neutrally) place a notification of this discussion at WT:WPM, although this was just started, so it might also be reasonable to wait a few days first to see if anyone else watching this weighs in. –Deacon Vorbis (carbon • videos) 18:01, 18 July 2020 (UTC)- I am sorry, I made another mistake, I did not want to ask for a poll, let me try again to build consensus on this. The method of partial fractions involves writing a proper (i.e the degree of the top less than the degree of the bottom) rational fraction as a sum of proper rational fractions (called partial fractions) with the bottom a power of an irreducible factor of the bottom of the original fraction, and unknown constants appearing as coefficients on the top, then determining the constants after adding the partial fractions, identifying the coefficients on the top of the two sides, and solving a system of linear equations. The proof that the method always works involves arithmetic in the polynomial ring (the euclidean and division algorithms), and is usually not done in calculus. The method of partial fractions is usually taught after trigonometric integrals and substitutions (see examples above), and so if somebody wants to teach the integral of secant using the Barrow or Weierstrass methods they have to wait until partial fractions are covered. However, integrating the particular fraction 1/(1-u^2) appearing in the two methods does not require knowing the general method of partial fractions, it can be integrated by simply adding and subtracting u in the top and separating the fractions, as shown in my edit. The decomposition obtained this way is 1/(1-u^2)=1/(1_u)+u/(1-u^2), which is not a partial fraction decomposition, because of the 1-u^2 denominator of the second fraction. Nevertheless, both fractions can be easily integrated, and the final result is the same as the one obtained by the method of partial fractions. My edit is therefore potentially useful, it allows people to teach the Barrows or Weierstrass methods before teaching the general method of partial fractions, if they so choose. Another (unintended) benefit of my edit is to make people aware that "partial fractions decomposition" has a precise meaning, writing a fraction as a sum of two other fractions is not generally a partial fraction decomposition.Sraianu (talk) 19:00, 18 July 2020 (UTC)
- The method of partial fraction decomposition can be taught at the level of Pre-calculus; I remember seeing this section right after the one for long division. Strictly, one can teach even at the level of Algebra II, if the students prove to be adept at symbolic manipulations. While it is true that in Calculus (either as a collegiate course or as AP Calculus BC), the method of partial fraction decomposition is commonly taught in the second semester and after integration trigonometric functions, I really do not see the point of adding such a minor change to the article, given the Barrow's approach of making a substitution. Integration by substitution is the first technique of integral evaluation students learned. On the other hand, the Weierstrass substitution is not commonly taught these days, only mentioned in some textbooks and perhaps lecture notes. In any case, given that the whole point of the Weierstrass substitution is to convert a rational function of sines and cosines into a rational function of polynomials, it makes no sense to teach or learn it without first being familiar with partial fraction decomposition (and long division). Nerd271 (talk) 20:21, 18 July 2020 (UTC)
- The whole integral calculus can be taught at the differential calculus level, and some of it even below that, it's just a matter on not having enough time to do it. My edit only shows why partial fractions are not needed to integrate 1/(1-u^2), I would be happy to have the edit restricted to just that, it would be enough for people for which this means something, and small enough so that it does not bother the rest. Again, the method of partial fractions has a precise definition. To see why this matters let's consider the expression "rational function of polynomials", which is pleonastic, a rational fraction (function) is by definition a fraction of polynomials (polynomial functions). As to the confusion between polynomials and polynomial functions, or between rational fractions and rational functions, people can get away with not knowing what the difference is because all fields appearing in calculus and below are infinite. Sraianu (talk) 20:56, 18 July 2020 (UTC)
- The method of partial fraction decomposition can be taught at the level of Pre-calculus; I remember seeing this section right after the one for long division. Strictly, one can teach even at the level of Algebra II, if the students prove to be adept at symbolic manipulations. While it is true that in Calculus (either as a collegiate course or as AP Calculus BC), the method of partial fraction decomposition is commonly taught in the second semester and after integration trigonometric functions, I really do not see the point of adding such a minor change to the article, given the Barrow's approach of making a substitution. Integration by substitution is the first technique of integral evaluation students learned. On the other hand, the Weierstrass substitution is not commonly taught these days, only mentioned in some textbooks and perhaps lecture notes. In any case, given that the whole point of the Weierstrass substitution is to convert a rational function of sines and cosines into a rational function of polynomials, it makes no sense to teach or learn it without first being familiar with partial fraction decomposition (and long division). Nerd271 (talk) 20:21, 18 July 2020 (UTC)
- I am sorry, I made another mistake, I did not want to ask for a poll, let me try again to build consensus on this. The method of partial fractions involves writing a proper (i.e the degree of the top less than the degree of the bottom) rational fraction as a sum of proper rational fractions (called partial fractions) with the bottom a power of an irreducible factor of the bottom of the original fraction, and unknown constants appearing as coefficients on the top, then determining the constants after adding the partial fractions, identifying the coefficients on the top of the two sides, and solving a system of linear equations. The proof that the method always works involves arithmetic in the polynomial ring (the euclidean and division algorithms), and is usually not done in calculus. The method of partial fractions is usually taught after trigonometric integrals and substitutions (see examples above), and so if somebody wants to teach the integral of secant using the Barrow or Weierstrass methods they have to wait until partial fractions are covered. However, integrating the particular fraction 1/(1-u^2) appearing in the two methods does not require knowing the general method of partial fractions, it can be integrated by simply adding and subtracting u in the top and separating the fractions, as shown in my edit. The decomposition obtained this way is 1/(1-u^2)=1/(1_u)+u/(1-u^2), which is not a partial fraction decomposition, because of the 1-u^2 denominator of the second fraction. Nevertheless, both fractions can be easily integrated, and the final result is the same as the one obtained by the method of partial fractions. My edit is therefore potentially useful, it allows people to teach the Barrows or Weierstrass methods before teaching the general method of partial fractions, if they so choose. Another (unintended) benefit of my edit is to make people aware that "partial fractions decomposition" has a precise meaning, writing a fraction as a sum of two other fractions is not generally a partial fraction decomposition.Sraianu (talk) 19:00, 18 July 2020 (UTC)
@Nerd271: Sorry, in my previous post I wanted to say rings, not fields. You are correct about the Weierstrass method, this is why when I teach integral calculus I only mention it as a general method for evaluating trigonometric integrals after I cover partial fractions, without even giving examples. But when I teach the integral of secant I usually do both the standard method and Barrow's method, the latter being my favorite, even though it's a bit longer.Sraianu (talk) 21:20, 18 July 2020 (UTC)
@Deacon Vorbis: and @Nerd271: I definitely do not plan to involve other users, my goal is to convince you two, if I fail it just means that I have failed, and that will be it. I apologize for the excessive communication. Sraianu (talk) 21:56, 18 July 2020 (UTC)
Integral of the hyperbolic secant
[edit]Anyone want to try making an article about the hyperbolic secant integral?