Lijst van integralen van goniometrische functies

Dit artikel bevat een lijst van integralen van goniometrische functies. Goniometrische functies zijn in de goniometrie gedefinieerde functies. Integralen zijn het onderwerp van studie van de integraalrekening.

Integralen met alleen een sinus

$\int \sin axdx=-\frac{1}{a}\cos ax+C$

$\int {\sin }^{2}axdx=\frac{x}{2}-\frac{1}{4a}\sin 2ax+C=\frac{x}{2}-\frac{1}{2a}\sin ax\cos ax+C$

$\int x{\sin }^{2}axdx=\frac{x^2}{4}-\frac{x}{4a}\sin 2ax-\frac{1}{8a^2}\cos 2ax+C$

$\int x^2{\sin }^{2}axdx=\frac{x^3}{6}-(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax-\frac{x}{4a^2}\cos 2ax+C$

$\int \sin b_{1}x\sin b_{2}xdx=\frac{\sin ((b_1-b_2)x)}{2(b_1-b_2)}-\frac{\sin ((b_1+b_2)x)}{2(b_1+b_2)}+C\text{(voor }|b_1|\ne |b_2|\text{)}$

$\int {\sin }^{n}axdx=-\frac{{\sin }^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int {\sin }^{n-2}axdx\text{(voor }n>0\text{)}$

$\int \frac{dx}{\sin ax}=\frac{1}{a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{dx}{{\sin }^{n}ax}=\frac{\cos ax}{a(1-n){\sin }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}ax}\text{(voor }n>1\text{)}$

$\int x\sin axdx=\frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C$

$\int x^n\sin axdx=-\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos axdx\text{(voor }n>0\text{)}$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\sin }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(voor }n=2,4,6...\text{)}$

$\int \frac{\sin ax}{x}dx=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{(ax{)}^{2n+1}}{(2n+1)\cdot (2n+1)!}+C$

$\int \frac{\sin ax}{x^n}dx=-\frac{\sin ax}{(n-1)x^{n-1}}+\frac{a}{n-1}\int \frac{\cos ax}{x^{n-1}}dx$

$\int \frac{dx}{1\pm \sin ax}=\frac{1}{a}\tan (\frac{ax}{2}\mp \frac{\pi }{4})+C$

$\int \frac{xdx}{1+\sin ax}=\frac{x}{a}\tan (\frac{ax}{2}-\frac{\pi }{4})+\frac{2}{a^2}\ln |\cos (\frac{ax}{2}-\frac{\pi }{4})|+C$

$\int \frac{xdx}{1-\sin ax}=\frac{x}{a}\cot (\frac{\pi }{4}-\frac{ax}{2})+\frac{2}{a^2}\ln |\sin (\frac{\pi }{4}-\frac{ax}{2})|+C$

$\int \frac{\sin axdx}{1\pm \sin ax}=\pm x+\frac{1}{a}\tan (\frac{\pi }{4}\mp \frac{ax}{2})+C$

Integralen met alleen een cosinus

$\int \cos axdx=\frac{1}{a}\sin ax+C$

$\int {\cos }^{n}axdx=\frac{{\cos }^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int {\cos }^{n-2}axdx\text{(voor }n>0\text{)}$

$\int x\cos axdx=\frac{\cos ax}{a^2}+\frac{x\sin ax}{a}+C$

$\int {\cos }^{2}axdx=\frac{x}{2}+\frac{1}{4a}\sin 2ax+C=\frac{x}{2}+\frac{1}{2a}\sin ax\cos ax+C$

$\int x^2{\cos }^{2}axdx=\frac{x^3}{6}+(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax+\frac{x}{4a^2}\cos 2ax+C$

$\int x^n\cos axdx=\frac{x^n\sin ax}{a}-\frac{n}{a}\int x^{n-1}\sin axdx$

$\int \frac{\cos ax}{x}dx=\ln |ax|+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{(ax{)}^{2k}}{2k\cdot (2k)!}+C$

$\int \frac{\cos ax}{x^n}dx=-\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int \frac{\sin ax}{x^{n-1}}dx\text{(voor }n\ne 1\text{)}$

$\int \frac{dx}{\cos ax}=\frac{1}{a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{dx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(voor }n>1\text{)}$

$\int \frac{dx}{1+\cos ax}=\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{dx}{1-\cos ax}=-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \frac{xdx}{1+\cos ax}=\frac{x}{a}\tan \frac{ax}{2}+\frac{2}{a^2}\ln |\cos \frac{ax}{2}|+C$

$\int \frac{xdx}{1-\cos ax}=-\frac{x}{a}\cot \frac{ax}{2}+\frac{2}{a^2}\ln |\sin \frac{ax}{2}|+C$

$\int \frac{\cos axdx}{1+\cos ax}=x-\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{\cos axdx}{1-\cos ax}=-x-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \cos a_{1}x\cos a_{2}xdx=\frac{\sin (a_1-a_2)x}{2(a_1-a_2)}+\frac{\sin (a_1+a_2)x}{2(a_1+a_2)}+C\text{(voor }|a_1|\ne |a_2|\text{)}$

$\int \sqrt{1+\cos x}dx=\sqrt{2}\int \cos \frac{1}{2}xdx=2\sqrt{2}\sin \frac{1}{2}x\text{overgaan naar de halve hoek}$

Integralen met alleen een tangens

$\int \tan axdx=-\frac{1}{a}\ln |\cos ax|+C=\frac{1}{a}\ln |\sec ax|+C$

$\int {\tan }^{n}axdx=\frac{1}{a(n-1)}{\tan }^{n-1}ax-\int {\tan }^{n-2}axdx\text{(voor }n\ne 1\text{)}$

$\int \frac{dx}{q\tan ax+p}=\frac{1}{p^2+q^2}(px+\frac{q}{a}\ln |q\sin ax+p\cos ax|)+C\text{(voor }{p}^2+q^2\ne 0\text{)}$

$\int \frac{dx}{\tan ax}=\frac{1}{a}\ln |\sin ax|+C$

$\int \frac{dx}{\tan ax+1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{dx}{\tan ax-1}=-\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\tan axdx}{\tan ax+1}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\tan axdx}{\tan ax-1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

Integralen met alleen een cotangens

$\int \cot axdx=\frac{1}{a}\ln |\sin ax|+C$

$\int {\cot }^{n}axdx=-\frac{1}{a(n-1)}{\cot }^{n-1}ax-\int {\cot }^{n-2}axdx\text{(voor }n\ne 1\text{)}$

$\int \frac{dx}{1+\cot ax}=\int \frac{\tan axdx}{\tan ax+1}$

$\int \frac{dx}{1-\cot ax}=\int \frac{\tan axdx}{\tan ax-1}$

Integralen met alleen een secans

$\int \sec axdx=\frac{1}{a}\ln |\sec ax+\tan ax|+C$

$\int {\sec }^{2}xdx=tanx+C$

$\int {\sec }^{n}axdx=\frac{{\sec }^{n-1}ax\sin ax}{a(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}axdx\text{ (voor }n\ne 1\text{)}$

$\int {\sec }^{n}xdx=\frac{{\sec }^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int {\sec }^{n-2}xdx$

$\int \frac{dx}{\sec x+1}=x-\tan \frac{x}{2}+C$

$\int \frac{dx}{\sec x-1}=-x-\cot \frac{x}{2}+C$

Integralen met alleen een cosecans

$\int \csc axdx=\frac{1}{a}\ln |\csc ax-\cot ax|+C$

$\int {\csc }^{2}xdx=-\cot x+C$

$\int {\csc }^{n}axdx=-\frac{{\csc }^{n-1}ax\cos ax}{a(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}axdx\text{ (voor }n\ne 1\text{)}$

$\int \frac{dx}{\csc x+1}=x-\frac{2\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}}+C$

$\int \frac{dx}{\csc x-1}=\frac{2\sin \frac{x}{2}}{\cos \frac{x}{2}-\sin \frac{x}{2}}-x+C$

Integralen met sinus en cosinus

$\int \frac{dx}{\cos ax\pm \sin ax}=\frac{1}{a\sqrt{2}}\ln |\tan (\frac{ax}{2}\pm \frac{\pi }{8})|+C$

$\int \frac{dx}{(\cos ax\pm \sin ax{)}^2}=\frac{1}{2a}\tan (ax\mp \frac{\pi }{4})+C$

$\int \frac{dx}{(\cos x+\sin x{)}^n}=\frac{1}{n-1}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}-2(n-2)\int \frac{dx}{(\cos x+\sin x{)}^{n-2}})$

$\int \frac{\cos axdx}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\cos axdx}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\sin axdx}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\cos axdx}{\sin ax(1+\cos ax)}=-\frac{1}{4a}{\tan }^2\frac{ax}{2}+\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\cos axdx}{\sin ax(1+-\cos ax)}=-\frac{1}{4a}{\cot }^2\frac{ax}{2}-\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\sin axdx}{\cos ax(1+\sin ax)}=\frac{1}{4a}{\cot }^2(\frac{ax}{2}+\frac{\pi }{4})+\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{\sin axdx}{\cos ax(1-\sin ax)}=\frac{1}{4a}{\tan }^2(\frac{ax}{2}+\frac{\pi }{4})-\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \sin ax\cos axdx=\frac{1}{2a}{\sin }^{2}ax+C$

$\int \sin a_{1}x\cos a_{2}xdx=-\frac{\cos (a_1-a_2)x}{2(a_1-a_2)}-\frac{\cos (a_1+a_2)x}{2(a_1+a_2)}+C\text{(voor }|a_1|\ne |a_2|\text{)}$

$\int {\sin }^{n}ax\cos axdx=\frac{1}{a(n+1)}{\sin }^{n+1}ax+C\text{(voor }n\ne -1\text{)}$

$\int \sin ax{\cos }^{n}axdx=-\frac{1}{a(n+1)}{\cos }^{n+1}ax+C\text{(voor }n\ne -1\text{)}$

$\int {\sin }^{2}ax{\cos }^{2}axdx=\frac{x}{8}-\frac{\sin 4ax}{32a}+C$

$\int {\sin }^{n}ax{\cos }^{m}axdx=-\frac{{\sin }^{n-1}ax{\cos }^{m+1}ax}{a(n+m)}+\frac{n-1}{n+m}\int {\sin }^{n-2}ax{\cos }^{m}axdx\text{(voor }m,n>0\text{)}$

$\int {\sin }^{n}ax{\cos }^{m}axdx=\frac{{\sin }^{n+1}ax{\cos }^{m-1}ax}{a(n+m)}+\frac{m-1}{n+m}\int {\sin }^{n}ax{\cos }^{m-2}axdx\text{(voor }m,n>0\text{)}$

$\int \frac{dx}{\sin ax\cos ax}=\frac{1}{a}\ln |\tan ax|+C$

$\int \frac{dx}{\sin ax{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+\int \frac{dx}{\sin ax{\cos }^{n-2}ax}\text{(voor }n\ne 1\text{)}$

$\int \frac{dx}{{\sin }^{n}ax\cos ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+\int \frac{dx}{{\sin }^{n-2}ax\cos ax}\text{(voor }n\ne 1\text{)}$

$\int \frac{\sin axdx}{{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+C\text{(voor }n\ne 1\text{)}$

$\int \frac{{\sin }^{2}axdx}{\cos ax}=-\frac{1}{a}\sin ax+\frac{1}{a}\ln |\tan (\frac{\pi }{4}+\frac{ax}{2})|+C$

$\int \frac{{\sin }^{2}axdx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(voor }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{\cos ax}=-\frac{{\sin }^{n-1}ax}{a(n-1)}+\int \frac{{\sin }^{n-2}axdx}{\cos ax}\text{(voor }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\frac{{\sin }^{n+1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}axdx}{{\cos }^{m-2}ax}\text{(voor }m\ne 1\text{)}$

$\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=-\frac{{\sin }^{n-1}ax}{a(n-m){\cos }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m}ax}\text{(voor }m\ne n\text{)}$

$\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\frac{{\sin }^{n-1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m-2}ax}\text{(voor }m\ne 1\text{)}$

$\int \frac{\cos axdx}{{\sin }^{n}ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+C\text{(voor }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}axdx}{\sin ax}=\frac{1}{a}(\cos ax+\ln |\tan \frac{ax}{2}|)+C$

$\int \frac{{\cos }^{2}axdx}{{\sin }^{n}ax}=-\frac{1}{n-1}(\frac{\cos ax}{a{\sin }^{n-1}ax)}+\int \frac{dx}{{\sin }^{n-2}ax})\text{(voor }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=-\frac{{\cos }^{n+1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\cos }^{n}axdx}{{\sin }^{m-2}ax}\text{(voor }m\ne 1\text{)}$

$\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=\frac{{\cos }^{n-1}ax}{a(n-m){\sin }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m}ax}\text{(voor }m\ne n\text{)}$

$\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=-\frac{{\cos }^{n-1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m-2}ax}\text{(voor }m\ne 1\text{)}$

Integralen met sinus en tangens

$\int \sin ax\tan axdx=\frac{1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C$

$\int {\sin }^{2}ax{\tan }^{2}axdx=\frac{\sin 2ax}{4a}-\frac{\tan ax}{a}-\frac{3x}{2}+C$

$\int \frac{{\tan }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n-1)}{\tan }^{n-1}(ax)+C\text{(voor }n\ne 1\text{)}$

Integralen met cosinus en tangens

$\int \frac{{\tan }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(n+1)}{\tan }^{n+1}ax+C\text{(voor }n\ne -1\text{)}$

Integralen met sinus en cotangens

$\int \frac{{\cot }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n+1)}{\cot }^{n+1}ax+C\text{(voor }n\ne -1\text{)}$

Integralen met cosinus en cotangens

$\int \frac{{\cot }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(1-n)}{\tan }^{1-n}ax+C\text{(voor }n\ne 1\text{)}$

Goniometrische integralen met symmetrische grenzen

$\underset{-c}{\overset{c}{\int }}\sin xdx=0$

$\underset{-c}{\overset{c}{\int }}\cos xdx=2\underset{0}{\overset{c}{\int }}\cos xdx=2\underset{-c}{\overset{0}{\int }}\cos xdx=2\sin c$

$\underset{-c}{\overset{c}{\int }}\tan xdx=0$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\cos }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(voor }n=1,3,5...\text{)}$

Literatuur

• Sjabloon:Aut. Calculus: Early Transcendentals, 2008.