Talk:List of integrals of irrational functions

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 $R^{1/2}$ 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 $R$s 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 $R$ ($R^{3}$, $R^{5}$ and $R^{2n+1}$). 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)

References

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

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)

Following your request I added to article on rational function a subdefinition about irrational function. Gradshteyn and Ryzhik do not use the term irrational function. Instead they use algebraic functions (roots or rational functions), trigonometric functions, exponential functions, logarithmic functions and so on. Perhaps one should to back to the definition of irrational function (which has a redirecto to rational functions) and add there these definitions. TomyDuby (talk) 19:12, 31 July 2008 (UTC)

Irrational function is somewhat unspecific. Abramowitz & Stegun use the term irrational algebraic functions. --catslash (talk) 23:19, 4 August 2008 (UTC)

Peirce B.O., Chap 3. is also called irrational algebraic functions --catslash (talk) 23:40, 6 August 2008 (UTC)

Propose move to List of integrals of irrational algebraic functions

Following the points raised in the preceding section, namely

1. irrational functions is not precise enough (it could include transcendental functions such as trigonometric functions)
2. various sources use the term irrational algebraic functions

I suggest moving the article to List of integrals of irrational algebraic functions. Any opinions? --catslash (talk) 22:09, 17 September 2008 (UTC)

The literature overwhelmingly calls these algebraic functions. In the spirit of WP:BOLD, I am going to move this to List of integrals of algebraic functions soon if there are no objections.

NB: Rational Functions even states "...the adjective "irrational" is not generally used for functions, but only for real numbers".

Asmeurer (talk ♬ contribs) 02:14, 11 June 2010 (UTC)

Just algebraic would be an improvement over just irrational, however I would prefer to keep both adjectives because:

• Rational functions are excluded; they have their own page at List of integrals of rational functions
• The only two sources I have to hand (Abramowitz & Stegun and Peirce B.O.) likewise have sections called Integrals of Rational Algebraic Functions and Integrals of Irrational Algebraic Functions (identical titles in the two publications). I'm not sure what literature you are referring to, but could you check that when it refers to algebraic functions, it isn't taken to include rational functions?

It's remarkable how little argument this topic has generated since Michael Hardy first questioned the title 3.5 years ago! --catslash (talk) 14:42, 11 June 2010 (UTC)

Different integration constants

Section 1, Line 1 has been changed by 133.1.207.152 from

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

to

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

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)

The same issue for (and eventually others)

$\int {\frac {dx}{r}},\qquad \int {\frac {x^{2}}{r}}\;dx$

129.104.29.1 (talk) 08:56, 24 October 2014 (UTC)

Yes it's true that

$\int {\frac {1}{r}}dx=\ln(x+r)$

to within a constant of integration, but we also want to say that

$\int {\frac {1}{r}}dx=\mathrm {arcsinh} \left({\frac {x}{a}}\right)$

which differs from the preceding by a constant of $\scriptstyle {\ln(a)}$, i.e.

${\begin{aligned}\int {\frac {1}{r}}dx&=\mathrm {arcsinh} \left({\frac {x}{a}}\right)\\&=\ln(x+r)-\ln(a)\\&=\ln \left({\frac {x+r}{a}}\right)\end{aligned}}$

--catslash (talk) 16:07, 24 October 2014 (UTC)

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)

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)

Mistake

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

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

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

You're clearly correct, since by differentiation (using $S={\sqrt {ax+b}}$ and $\mathrm {\frac {\partial }{\mathrm {\partial } x}} S={\frac {a}{2S}}$),

${\begin{aligned}\mathrm {\frac {\partial }{\mathrm {\partial } x}} \left({\frac {2}{a(2n+3)}}(x^{n}S^{3}-nb\int x^{n-1}S\mathrm {\partial } x)\right)&={\frac {2}{a(2n+3)}}\left(nx^{n-1}S^{3}+3x^{n}S{\frac {a}{2}}-nb\mathrm {\frac {\partial }{\mathrm {\partial } x}} \int x^{n-1}S\mathrm {\partial } x\right)\\&={\frac {2}{a(2n+3)}}\left(nx^{n-1}S^{3}+3x^{n}S{\frac {a}{2}}-nbx^{n-1}S\right)\\&={\frac {2x^{n-1}S}{a(2n+3)}}\left(nax+3x{\frac {a}{2}}\right)\\&=x^{n}S\end{aligned}}$

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)

Another mistake?

Consider $\int {dx \over {xR}}=-{1 \over {\sqrt {c}}}arsinh\left({{bx+c} \over {\left|x\right|{\sqrt {4ac-b^{2}}}}}\right),c<0$

The answer is imaginary; I can't see how that can be the intent. I don't have time to check it immediately, but I suspect ${\sqrt {c}}$ should be ${\sqrt {-c}}$ and the condition should be $c<0,4ac-b^{2}>0$ .197.88.91.114 (talk) 06:20, 5 April 2015 (UTC)

The integral is correct as written (certainly to within a sign), it's not out by a factor i - it's just the condition which is inappropriate. --catslash (talk) 17:45, 5 April 2015 (UTC)