Talk:List of integrals of irrational functions

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[edit] Confusion

There is some ambiguity in the fourth and fifth sections, what exactly is R defined as? --Monguin61 03:45, 10 December 2005 (UTC)

I have added a flag on the page warning of this. --Laura, 19 January 2007

I reckon;
  • Section 4, line 8; \sqrt{R} is used where R was intended - and probably on some of the other lines too.
  • Section 5 redefines R - I'm going to change this to S making it consistent with the preceding sections.
  • Section 5 defines the R1 / 2 rather than R (soon to be S) as equal to the radical, which is inconsistent with the previous sections.
  • Sections 4 and 5 don't refer to R in the integrand, which is inconsistent with the previous sections.
--catslash 01:03, 22 January 2007 (UTC)
If you go to the Russian version of the page, section 4 is entitled Интегралы с R^{1/2} = \sqrt{ax^2+bx+c}, and not R = \sqrt{ax^2+bx+c} (I should have thought of looking there earlier). Then, except for the first integral in the section, the Rs appear as they did on the English page. Assuming the Russians are correct (which seems likely), the options are (1) press on changing \sqrt{R} to R, or (2) revert to \sqrt{R} and then change the section title and the first integral accordingly. Any opinions? (I'm for the former option). --catslash 02:28, 22 January 2007 (UTC)
The usage of R in section 4 is now self-consistent: it's what was designated \sqrt{R} originally, before it got confused. I don't suppose it matters that much, whether \sqrt{ax^2+bx+c} is called \sqrt{R} or R, but the latter (present) usage is (1) more consistent with the preceding sections and (2) simpler, since no even powers of \sqrt{ax^2+bx+c} ever occur.
Abramowitz and Stegun verifies all the integrals in section 4 which don't involve higher powers of R (R3, R5 and R2n + 1). I've also verified all the integrals in section 4 by direct differentiation. Consequently, I thought it would be OK to remove the warning tag on this section. However, we should be able to cite sources for all the integrals given, so I shall add a different tag at the bottom of the page. --catslash 23:30, 28 January 2007 (UTC)

[edit] References

This page needs to cite its references for these integrals. --Pierremenard 19:48, 24 January 2006 (UTC)

[edit] Definition?

What exactly does the term irrational function mean? Perhaps one could define it as any function that is not a rational function, but by usage it seems to mean square roots of rational functions. Is that what was intended? Maybe that should be stated explicitly in the article. Michael Hardy 21:57, 10 November 2006 (UTC)

[edit] Different integration constants

Section 1, Line 1 has been changed by 133.1.207.152 from

\int r \;dx = \frac{1}{2}\left(x r +a^2\,\ln\left(x+r\right)\right)

to

\int r \;dx = \frac{1}{2}\left(x r +a^2\,\ln\left(\frac{x+r}{a}\right)\right)

These are both equally correct; it's just that the constant of integration differs by \frac{1}{2} a^{2} \mathrm{ln}(a). However, the former expression seems simpler, so I'm going to change it back to how it was. If there's some good reason for preferring the latter expression, then let us know, and I'd be happy to go with that one. --catslash 14:18, 2 February 2007 (UTC)

[edit] Simbols are inconsistent

I noticed that to denote the function 'arcsin' it is sometimes used the symbol 'sin-1' (as in the first example of the "integrals involving t = sqrt(a2-x2)"), and sometimes the symbol 'arcsin' (4th example of the following subsection). We should agree on one symbol and be consistent with it.

I suggest that 'arcsin' should be used, as 'sin-1' may also mean '1/sin'

cheers

Paolothecurious 04:45, 9 February 2007 (UTC)

Agreed: I've just changed three sin − 1 -> arcsin, but there's a cos − 1 and plenty of sinh − 1 and tanh − 1 left to do.--catslash 01:10, 12 February 2007 (UTC)

[edit] Definite integrals

Where can I find a list of definite integrals? For example, in the Student's t-distribution page, we have a formula for the probability density function, from which we may infer that \int\limits_{-\infty}^{\infty} (1 + x^2 / \nu)^{-(\nu + 1)/2} dx\, is some pretty expression involving the Gamma function. But where can I find it? Albmont 21:07, 29 March 2007 (UTC)

[edit] Mistake

The last integral of the table ist incorrect. The correct solution is
\int x^n \sqrt{ax + b}\,dx \; = \; \frac{2}{2n +3}\left(x^{n+1} \sqrt{ax + b} + \frac ba x^{n} \sqrt{ax + b} - n\frac ba \int x^{n-1}\sqrt{ax + b}\,dx \right)

which should better be written as
\int x^n \sqrt{ax + b}\,dx \; = \; \frac{2}{a(2n+3)}\left(x^n(ax+b) \sqrt{ax + b} - nb\int x^{n-1}\sqrt{ax + b}\,\mathrm dx \right) \qquad\mbox{(}n\,\ge\,1\mbox{)}

-- Peter Steinberg (Germany) (talk) 15:52, 18 April 2008 (UTC)

You're clearly correct, since by differentiation (using S = \sqrt{a x + b} and \mathrm{\frac{\partial}{\mathrm{\partial}x}}S = \frac{a}{2 S}),
\begin{align}\mathrm{\frac{\partial}{\mathrm{\partial}x}}\left(\frac{2}{a (2 n + 3)} (x^{n} S^{3} - n b \int x^{n - 1} S \mathrm{\partial}x)\right) & = \frac{2}{a (2 n + 3)} \left( n x^{n - 1} S^{3} + 3 x^{n} S \frac{a}{2} - n b \mathrm{\frac{\partial}{\mathrm{\partial}x}}\int x^{n - 1} S \mathrm{\partial}x\right) \\ & = \frac{2}{a (2 n + 3)} \left( n x^{n - 1} S^{3} + 3 x^{n} S \frac{a}{2} - n b x^{n - 1} S\right) \\ & = \frac{2 x^{n - 1} S}{a (2 n + 3)} \left( n a x + 3 x \frac{a}{2}\right) \\ & = x^{n} S \end{align}
and the difference between your answer and the one currently given depends on x (so it doesn't merely differ by a constant of integration). I will correct the article. Did you just spot this, or do you have some source that we can reference? --catslash (talk) 18:47, 18 April 2008 (UTC)
As far as I can see, the only “source” you need ist the calculation you did just.
If you are want know how I found it out: I'm working on a list of integrals for German “wikibooks”. (You can find it by entering “Formelsammlung Mathematik: Integrale” at the this link.) One source, of course, is your list of integrals. Trying to prove your results by differentiation, I could easily “spot” what is wrong. -- Peter Steinberg (Germany) (talk) 23:07, 23 April 2008 (UTC)