Talk:List of integrals of trigonometric functions

Name of this article

Some suggestion. Shouldn't we name this like List of Integrals involving trigonometric functions or short, List of Integrals with trigonometric functions. Use of parentheses looks like disambiguation. -- Taku 15:24 27 May 2003 (UTC)

"List of integrals of trigonometric functions". And a lower case "i" for "integrals" too! -- Tarquin 15:30 27 May 2003 (UTC)

Either will be fine with me. -- schnee

The section titles here look kind of wordy and redundant. The user probably knows he's looking at integrals of trigonometric functions. You could probably get by with just 'Integrals containing only sin', etc. --67.137.24.115 01:11, 21 November 2006 (UTC)

Is there a mistake in $\int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{n-1}}\int {\frac {\sin ^{n-1}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!$? Especially in the ${\frac {n-1}{n-1}}$ part. --Telempe 10:35, 18 February 2007 (UTC)

Fixed --RDBury (talk) 03:47, 8 March 2008 (UTC)

Derivations

Shouldn't we put the derivations/proofs for at least the more basic of these formulas? Kr5t 22:58, 7 March 2007 (UTC)

+ C

Since these are all improper integrals, shouldn't each of these end in + C. It's not overly important, but text books still do it, and it would be technically correct, which is the best kind of correct. . . . 10:27, 30 June 2007 (UTC)

The $+c$ is important for consistency. For example integrating $2\cos x\sin x$ two different ways gives $\sin ^{2}x$ and $-\cos ^{2}x$, both are correct but without the $+c$ you get $\sin ^{2}x=-\cos ^{2}x$, which is wrong. I'm going to try to add the +c's and change the c's that are already there to a's to avoid collision. —Preceding unsigned comment added by RDBury (talk • contribs) 04:02, 8 March 2008 (UTC)

Removed cot/tan section

I removed the cot and tan section. The only integral in it trivially simplified to another integral in this page.--RDBury (talk) 06:31, 8 March 2008 (UTC)

Deletions

I'm removing

$\int {\sqrt {1-\sin {x}}}\,dx=\int {\sqrt {\operatorname {cvs} \,{x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}{\sqrt {\operatorname {cvs} \,{x}}}+c=2{\sqrt {1+\sin {x}}}+c$

First, there are no corresponding 1+sinx or 1-cosx integrals listed. Second, cvs is rarely used and appears nowhere else on the page. Third, the intermediate expressions are confusing, I'm not sure if it's supposed to be a derivation. Fourth, the integrand simplifies to an expression
(|cos x/2 - sin x/2|) which can be solved using other formulas in the page. —Preceding unsigned comment added by RDBury (talk • contribs) 17:43, 16 March 2008 (UTC)

Removing

$\int \sin x\,dx=-\cos x+c\,\!$

It's a trivial special case of the previous integral.--RDBury (talk) 17:49, 16 March 2008 (UTC)

Clutter

Most of the formulas, such as the first listed formula,

$\int \sin ax\;dx=-{\frac {1}{a}}\cos ax+C\,\!$

contain "$ax\!$" instead of just "$x\!$". These "$a\!$"'s are unnecessary and distracting clutter. All we need to do is say once that if

$\int f(x)dx=F(x)+C$

and $a\neq 0\!$, then it follows that

$\int f(ax)dx={\frac {1}{a}}F(ax)+C.$

After all, we do not include the limits of integration (another possible type of clutter) explicitly in most cases, do we? JRSpriggs (talk) 03:57, 12 July 2008 (UTC)

You may have a point. But, whether it makes sense or not, this does seem to be the style most used in standard integration tables (see e.g. Abramowitz and Stegun). Also, once you start allowing trivial substitutions, the question of where to draw the line between trivial and nontrivial arises. For example, do you change $\int {1 \over {a^{2}+x^{2}}}dx$ to $\int {1 \over {1+x^{2}}}dx$? Do you omit $\int \sin ^{2}x\;dx$ entirely because it reduces to $\int \cos 2x\;dx$? IMO It's better to defer to existing standards on this rather than start a lot of debate on what constitutes a trivial simplification.--RDBury (talk) 12:42, 12 July 2008 (UTC)

Error?

The expression $\int {\frac {\cos ^{n}ax\;dx}{\sin ^{m}ax}}=-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-m-2}{m-1}}\int {\frac {\cos ^{n}ax\;dx}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!$ contains a factor $(n-m-2)$. Shouldn't this be $(n-m+2)$? — Preceding unsigned comment added by 88.159.101.36 (talk) 16:07, 7 April 2012 (UTC)

Inconsistent Notation

This article uses sin(n-m)x to indicate sin[(n-m)x] in one context (see the last integral given under "Integrals involving only cosine") and cos((n-m)x) to indicate cos[(n-m)x] in another context (see "Integrals involving sine and cosine". I have not edited it because I'm not certain what the author means to denote in the section on "Sine and Cosine", but the notation in "Integrals involving only cosine" is misleading. If x is a element in a product and not an argument of the Sine function in that integral, it leads to non-zero Fourier coefficients that should be zero, so that notation can easily be read in a way that is misleading and wrong. — Preceding unsigned comment added by 98.223.176.69 (talk) 22:22, 28 May 2013 (UTC)

Integral over a full circle

In the formulae

• $\int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{n}{x}\,dx=0\!\qquad n,m\in \mathbb {Z}$
• $\int _{0}^{2\pi }\sin ^{m}{x}\cos ^{2n+1}{x}\,dx=0\!\qquad n,m\in \mathbb {Z}$

one finds $n,m\in \mathbb {Z}$. Really? I would suggest $n,m\in \mathbb {N}$ (including $n=0$ and $m=0$). Stefan Neumeier (talk) 09:55, 31 May 2024 (UTC)