Primitives de funcions trigonomètriques

Tot seguit es presenta una llista de les primitives (o integrals) de funcions trigonomètriques. Per a consultar les integrals que impliquen funcions exponencials i trigonomètriques, veure Llista d'integrals de funcions exponencials. Per a consultar una llista completa de primitives de tota mena de funcions adreceu-vos a taula d'integrals

En totes les fórmules, la constant a se suposa diferent de zero i C indica la constant d'integració.

Integrals de funcions trigonomètriques que inclouen només el sinus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrals de funcions trigonomètriques que inclouen només el cosinus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrals de funcions trigonomètriques que inclouen només la tangent

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

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

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

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

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

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

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

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

Integrals de funcions trigonomètriques que inclouen només la secant

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

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

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

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

Integrals de funcions trigonomètriques que inclouen només la cosecant

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

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

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

Integrals de funcions trigonomètriques que contenen només la cotangent

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

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

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

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

Integrals de funcions trigonomètriques que inclouen ambdós sinus i cosinus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrals de funcions trigonomètriques que inclouen ambdós sinus i tangent

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

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

Integrals de funcions trigonomètriques que inclouen ambdós cosinus i tangent

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

Integrals de funcions trigonomètriques que inclouen ambdós sinus i cotangent

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

Integrals de funcions trigonomètriques que inclouen ambdós cosinus i cotangent

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

Integrals de funcions trigonomètriques amb limits simètrics

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

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

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

Integral en un cercle complet

$\underset{0}{\overset{2\pi }{\int }}{\sin }^{2m+1}x{\cos }^{2n+1}xdx=0n,m\in \mathbb{Z}$