Khác biệt giữa các bản “Danh sách tích phân với hàm lượng giác”

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{{Danh sách tích phân}}
{{Lượng giác}}
Đây là danh sách các '''[[tích phân]] ([[nguyên hàm]]) của các [[hàm lượng giác]]'''. Đối với tích phân của hàm số cóchứa hàm lượng giác và hàm mũ, xem [[danhDanh sách tích phân với hàm mũ]]. Đối với danh sách đầy đủ các tích phân, xem [[danhDanh sách tích phân]]. XemĐối thêmvới danh sách các tích phân đặc biệt của các hàm lượng giác, xem [[tíchTích phân lượng giác]].
 
MộtNhìn cách tổng quátchung, với <math>\cos(x)</math> là đạo hàm của hàm số <math>\sin(x)</math>, ta có
 
: <math>\int a\cos nx\;,dx = \frac{a}{n}\sin nx+C</math>
Trong mọi công thức dưới đây, {{mvar|a}} là một hằng số khác không âm{{mvar|C}} ký hiệu củacho [[hằng số tích phân]].
 
== Tích phân chỉ chứa hàm [[sin]] ==
 
: <math>\int\sin ax\;,dx = -\frac{1}{a}\cos ax+C\,\!</math>
: <math>\int\sin^2 {ax}\;dx = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +C\!</math>
: <math>\int\sin^3 {ax}\;dx = \frac{\cos 3ax}{12a} - \frac{3 \cos ax}{4a} +C\!</math>
 
: <math>\int x\sin^2 {ax}\;,dx = \frac{x^2}{42} - \frac{x1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{8a^22a} \sin ax\cos 2axax +C\!</math>
 
: <math>\int x^2\sin^2 {ax}\;dx = \frac{x^3}{6} - \left(\frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C\!</math>
: <math>\int\sin^3 b_1x{ax}\sin b_2x\;,dx = \frac{\sin((b_1-b_2)x)cos 3ax}{2(b_1-b_2)12a} - \frac{3 \sin((b_1+b_2)x)cos ax}{2(b_1+b_2)4a} +C \qquad(|b_1|\neq|b_2|)\,\!</math>
 
: <math>\int\sin^n {ax}\;dx = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\;dx \qquad(n>2)\,\!</math>
: <math>\int\frac{dx}{ x\sin^2 {ax}\,dx = \frac{1x^2}{a4} - \lnfrac{x}{4a} \left|\tansin 2ax - \frac{ax1}{8a^2} \right|cos 2ax +C</math>
 
: <math>\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} \qquad(n>1)\,\!</math>
: <math>\int x^2\sin^2 {ax}\;,dx = \frac{x^3}{6} - \sinleft( \frac ax}{ax^2}{4a} - \frac{x1}{8a^3} \cosright) ax\sin 2ax - \frac{x}{a4a^2} \cos 2ax +C\,\!</math>
 
: <math>\int x^n\sin ax\;dx = -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\;dx = \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \qquad(n>0)\,\!</math>
: <math>\int_{\frac{-a}{2}}^{\frac{a}{2}}int x^2\sin^2 {ax\,dx = \frac{n\pisin xax}{a^2}}\;dx = -\frac{a^3(n^2x\pi^2-6)cos ax}{24n^2\pi^2a} \qquad(n=2,4,6...)\,\!+C</math>
 
: <math>\int\frac{\sin ax}{x} dx = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C\,\!</math>
: <math>\int(\frac{sin b_1x)(\sin ax}{x^n} b_2x)\,dx = -\frac{\sin ax}{(n(b_2-1b_1)x^)}{n2(b_2-1b_1)}} + -\frac{a\sin((b_1+b_2)x)}{n-12(b_1+b_2)}+C \intqquad\fracmbox{\cos ax(}{x^{n-1}} dx|b_1|\,neq|b_2|\!mbox{)}</math>
 
: <math>\int\sin^n {ax}\,dx = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\,dx \qquad\mbox{(}n>0\mbox{)}</math>
 
: <math>\int\frac{dx}{\sin ax} = -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C</math>
 
: <math>\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} \qquad\mbox{(}n>1\mbox{)}</math>
 
: <math>\begin{align}
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(}n>0\mbox{)}
\end{align}</math>
 
: <math>\int\frac{\sin ax}{x}\,dx = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C</math>
 
: <math>\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</math>
: <math>\int\frac{dx}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+C</math>
 
: <math>\int\frac{x\;dx}{1+\sin ax} = \frac{x}{a}\tan\left(\frac{ax}{2} - \frac{\pi}{4}\right)+\frac{2}{a^2}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}\right)\right|+C</math>
: <math>\int\frac{x\;,dx}{1-+\sin ax} = \frac{x}{a}\cottan\left(\frac{\piax}{42} - \frac{ax\pi}{24}\right)+\frac{2}{a^2}\ln\left|\sincos\left(\frac{\piax}{42}-\frac{ax\pi}{24}\right)\right|+C</math>
 
: <math>\int\frac{\sin ax\;dx}{1\pm\sin ax} = \pm x+\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)+C</math>
: <math>\int\frac{x\,dx}{1-\sin ax} = \frac{x}{a}\cot\left(\frac{\pi}{4} - \frac{ax}{2}\right)+\frac{2}{a^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right)\right|+C</math>
 
: <math>\int\frac{\sin ax\,dx}{1\pm\sin ax} = \pm x+\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)+C</math>
 
== Tích phân chỉ chứa hàm [[Hàm lượng giác|cos]] ==
 
: <math>\int\cos ax\;,dx = \frac{1}{a}\sin ax+C\,\!</math>
 
: <math>\int\cos^2 {ax}\;,dx = \frac{x}{2} + \frac{1}{4a} \sin 2ax +C = \frac{x}{2} + \frac{1}{2a} \sin ax\cos ax +C\!</math>
 
: <math>\int\cos^n ax\;,dx = \frac{\cos^{n-1} ax\sin ax}{na} + \frac{n-1}{n}\int\cos^{n-2} ax\;,dx \qquad\mbox{(}n>0)\,\!mbox{)}</math>
 
: <math>\int x\cos ax\;,dx = \frac{\cos ax}{a^2} + \frac{x\sin ax}{a}+C\,\!</math>
 
: <math>\int x^2\cos^2 {ax}\;,dx = \frac{x^3}{6} + \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax + \frac{x}{4a^2} \cos 2ax +C\!</math>
 
: <math>\begin{align}
: <math>\int x^n\cos ax\;dx = \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\;dx\,= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \!</math>
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(}n>0\mbox{)}
\end{align}</math>
 
: <math>\int\frac{\cos ax}{x} \,dx = \ln|ax|+\sum_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}+C\,\!</math>
 
: <math>\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 \qquad\mbox{(}n\neq 1)\,\!mbox{)}</math>
 
: <math>\int\frac{dx}{\cos ax} = \frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C</math>
 
: <math>\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} \qquad\mbox{(}n>1)\,\!mbox{)}</math>
 
: <math>\int\frac{dx}{1+\cos ax} = \frac{1}{a}\tan\frac{ax}{2}+C\,\!</math>
 
: <math>\int\frac{dx}{1-\cos ax} = -\frac{1}{a}\cot\frac{ax}{2}+C\,\!</math>
 
: <math>\int\frac{x\;,dx}{1+\cos ax} = \frac{x}{a}\tan\frac{ax}{2} + \frac{2}{a^2}\ln\left|\cos\frac{ax}{2}\right|+C</math>
 
: <math>\int\frac{x\;,dx}{1-\cos ax} = -\frac{x}{a}\cot\frac{ax}{2}+\frac{2}{a^2}\ln\left|\sin\frac{ax}{2}\right|+C</math>
 
: <math>\int\frac{\cos ax\;,dx}{1+\cos ax} = x - \frac{1}{a}\tan\frac{ax}{2}+C\,\!</math>
 
: <math>\int\frac{\cos ax\;,dx}{1-\cos ax} = -x-\frac{1}{a}\cot\frac{ax}{2}+C\,\!</math>
 
: <math>\int(\cos a_1x)(\cos a_2x)\;,dx = \frac{\sin(a_1-(a_2-a_1)x)}{2(a_1-a_2-a_1)}+\frac{\sin(a_1+(a_2+a_1)x)}{2(a_1+a_2+a_1)}+C \qquad\mbox{(}|a_1|\neq|a_2|)\,\!mbox{)}</math>
 
== Tích phân chỉ chứa hàm [[hàm lượng giác|tan]] ==
 
: <math>\int\tan ax\;,dx = -\frac{1}{a}\ln|\cos ax|+C = \frac{1}{a}\ln|\sec ax|+C\,\!</math>
 
: <math>\int \tan^n2{x} ax\;, dx = \frac{1}{a(n-1)}\tan^{n-1x} ax-\int\tan^{n-2} ax\;dxx \qquad(n\neq 1)\,\!+C</math>
 
: <math>\int\tan^n ax\,dx = \frac{1}{a(n-1)}\tan^{n-1} ax-\int\tan^{n-2} ax\,dx \qquad(n\neq 1)\,\!</math>
 
: <math>\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 \qquad(p^2 + q^2\neq 0)\,\!</math>
: <math>\int\frac{dx}{\tan ax - 1} = -\frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C\,\!</math>
 
: <math>\int\frac{\tan ax\;,dx}{\tan ax + 1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax + \cos ax|+C\,\!</math>
 
: <math>\int\frac{\tan ax\;,dx}{\tan ax - 1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C\,\!</math>
 
== Tích phân chỉ chứa hàm [[hàm lượng giác|secant]] ==
: ''Xem [[{{more|Tích phân của hàm secant]].''}}
 
:<math>\int \sec{ax} \, dx = \frac{1}{a}\ln{\left| \sec{ax} + \tan{ax}\right|}+C</math>
:<math>\int \sec^2{x} \, dx = \tan{x}+C</math>
 
:<math>\int \sec^n{ax} \, dx = \frac{\sec^{n-2}{ax} \tan {ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{ax} \, dx \qquad \mbox{(}n\ne 1\mbox{)}\,\!</math>
 
:<math>\int \sec^n{x} \, dx = \frac{\sec^{n-2}{x}\tan{x}}{n-1} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{x}\,dx</math><ref>Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008</ref>
 
:<math>\int \frac{dx}{\sec{x} - 1} = - x - \cot{\frac{x}{2}}+C</math>
<!--
In the 17th century, the integral of the secant function was the subject of a well-known conjecture posed in the 1640s by Henry Bond. The problem was solved by [[Isaac Barrow]].<ref>V. Frederick Rickey and Philip M. Tuchinsky, "An Application of Geography to Mathematics: History of the Integral of the Secant", ''[[Mathematics Magazine]]'', volume 53, number 3, May 2980, pages 162–166</ref> It was originally for the purposes of [[cartography]] that this was needed. -->
 
== Tích phân chỉ chứa hàm [[hàm lượng giác|cosecant]] ==
 
:<math>\begin{align}
:<math>\int \csc{ax} \, dx = -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C</math>
\int \csc{ax} \, dx &= -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C\\
&= \frac{1}{a}\ln{\left| \csc{ax}-\cot{ax}\right|}+C\\
&= \frac{1}{a}\ln{\left| \tan{\left( \frac{ax}{2} \right)}\right|}+C
\end{align}</math>
 
:<math>\int \csc^2{x} \, dx = -\cot{x}+C</math>
 
:<math>\begin{align}
:<math>\int \csc^n{ax} \, dx = -\frac{\csc^{n-1}{ax} \csc{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, dx \qquad(n \ne 1)\,</math>
\int \csc^3{x} \, dx &= -\frac{1}{2}\csc x \cot x - \frac{1}{2}\ln|\csc x + \cot x| + C\\
&= -\frac{1}{2}\csc x \cot x + \frac{1}{2}\ln|\csc x - \cot x| + C
\end{align}</math>
 
:<math>\int \frac{dx}{\csc^n{xax} +\, 1}dx = x - \frac{2\sincsc^{\fracn-2}{xax} \cot{2}ax}}{a(n-1)} \cos{,+\, \frac{xn-2}{n-1}\int \csc^{n-2}{ax}+ \sin{, dx \fracqquad \mbox{x (}n \ne 1\mbox{2)}}}+C</math>
 
:<math>\int \frac{dx}{\csc{x} -+ 1} = x - \frac{2\sin{\frac{x}{2}}}{\coscot{\frac{x}{2}}-\sin{\frac{x+1}{2}}}-x+C</math>
 
:<math>\int \frac{dx}{\csc{x} - 1} = - x + \frac{2}{\cot{\frac{x}{2}}-1}+C</math>
 
== Tích phân chỉ chứa hàm [[hàm lượng giác|cotang]] ==
 
:<math>\int\cot ax\;,dx = \frac{1}{a}\ln|\sin ax|+C\,\!</math>
 
: <math>\int\cot^n ax\;dx = -\frac{1}{a(n-1)}\cot^{n-1} ax - \int\cot^{n-2} ax\;dx \qquad(n\neq 1)\,\!</math>
: <math>\int \fraccot^2{dxx}{1 + \cot, ax}dx = -\int\fraccot{\tan ax\;dxx}{\tan ax- x +1}\,\!C</math>
 
: <math>\int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\;dx}{\tan ax-1}\,\!</math>
: <math>\int\cot^n ax\,dx = -\frac{1}{a(n-1)}\cot^{n-1} ax - \int\cot^{n-2} ax\,dx \qquad\mbox{(}n\neq 1\mbox{)}</math>
 
: <math>\int\frac{dx}{1 + \cot ax} = \int\frac{\tan ax\,dx}{\tan ax+1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax + \cos ax|+C </math>
 
: <math>\int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\,dx}{\tan ax-1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C </math>
 
== Tích phân chứa hàm [[sin]] và [[hàm lượng giác|cos]] ==
Tích phân một hàm hữu tỉ (phân thức) của {{math|sin}} và {{math|cos}} có thể được tính bằng [[quy tắc Bioche]].
 
: <math>\int\frac{dx}{\cos ax\pm\sin ax} = \frac{1}{a\sqrt{2}}\ln\left|\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+C</math>
: <math>\int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right)</math>
 
: <math>\int\frac{\cos ax\;,dx}{\cos ax +\pm \sin ax} = \frac{x}{2} +\pm \frac{1}{2a}\ln\left|\sin ax +\pm \cos ax\right|+C</math>
 
: <math>\int\frac{\cossin ax\;,dx}{\cos ax -\pm \sin ax} = \pm\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax -\pm \cos ax\right|+C</math>
 
: <math>\int\frac{\sincos ax\;,dx}{(\cossin ax )(1+ \sincos ax)} = -\frac{x1}{4a}\tan^2\frac{ax} - {2}+\frac{1}{2a}\ln\left|\sin ax + tan\cos frac{ax}{2}\right|+C</math>
 
: <math>\int\frac{\sincos ax\;,dx}{(\cossin ax )(1- \sincos ax)} = -\frac{x1}{4a}\cot^2\frac{ax}{2} - \frac{1}{2a}\ln\left|\sin ax - tan\cos frac{ax}{2}\right|+C</math>
 
: <math>\int\frac{\cossin ax\;,dx}{(\sincos ax)(1+\cossin ax)} = -\frac{1}{4a}\tancot^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C</math>
 
: <math>\int\frac{\cossin ax\;,dx}{(\sincos ax)(1-\cossin ax)} = -\frac{1}{4a}\cottan^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C</math>
 
: <math>\int\frac{(\sin ax\;dx}{)(\cos ax(1+)\sin ax)},dx = \frac{1}{4a2a}\cotsin^2\left(\frac{ ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right| +C</math>
 
: <math>\int\frac{(\sin ax\;dx}{a_1x)(\cos ax(1-a_2x)\sin ax)},dx = -\frac{1}{4a}\tan^2\leftcos(\frac{ax(a_1-a_2)x)}{2(a_1-a_2)}+\frac{\pi}{4}\right) -\frac{1}{2a}\ln\left|\tan\leftcos(\frac{ax(a_1+a_2)x)}{2(a_1+a_2)} +C\frac{qquad\pi}mbox{4(}|a_1|\right)\rightneq|+Ca_2|\mbox{)}</math>
 
: <math>\int(\sin^n ax)(\cos ax)\;,dx = -\frac{1}{2aa(n+1)}\cossin^2{n+1} ax +C\,qquad\!mbox{(}n\neq -1\mbox{)}</math>
 
: <math>\int(\sin a_1xax)(\cos^n a_2xax)\;,dx = -\frac{\cos((a_1-a_2)x)1}{2a(a_1-a_2n+1)} -\frac{\cos((a_1+a_2)x)}^{2(a_1n+a_2)1} ax +C\qquad\mbox{(|a_1|}n\neq|a_2|) -1\,\!mbox{)}</math>
 
: <math>\begin{align}
: <math>\int\sin^n ax\cos ax\;dx = \frac{1}{a(n+1)}\sin^{n+1} ax +C\qquad(n\neq -1)\,\!</math>
\int(\sin^n ax)(\cos^m ax)\,dx &= -\frac{(\sin^{n-1} ax)(\cos^{m+1} ax)}{a(n+m)}+\frac{n-1}{n+m}\int(\sin^{n-2} ax)(\cos^m ax)\,dx \qquad\mbox{(}m,n>0\mbox{)} \\
&= \frac{(\sin^{n+1} ax)(\cos^{m-1} ax)}{a(n+m)} + \frac{m-1}{n+m}\int(\sin^n ax)(\cos^{m-2} ax)\,dx \qquad\mbox{(} m,n>0 \mbox{)}
\end{align}</math>
 
: <math>\int\frac{dx}{(\sin ax)(\cos^n ax\;dx)} = -\frac{1}{a(n+1)}\cos^{n+1}ln\left|\tan ax \right|+C\qquad(n\neq -1)\,\!</math>
 
: <math>\int\frac{dx}{(\sin^n ax)(\cos^mn ax\;dx)} = -\frac{\sin^1}{a(n-1} ax)\cos^{m+n-1} ax}{a(n+m)}+\int\frac{n-1dx}{n+m}\int(\sin ax)(\cos^{n-2} ax\cos^m ax\;dx)} \qquad\mbox{(m,}n>0)\,neq 1\!mbox{)}</math>
: và: <math>\int\sin^n ax\cos^m ax\;dx = \frac{\sin^{n+1} ax\cos^{m-1} ax}{a(n+m)} + \frac{m-1}{n+m}\int\sin^n ax\cos^{m-2} ax\;dx \qquad(m,n>0)\,\!</math>
 
: <math>\int\frac{dx}{(\sin^n ax)(\cos ax)} = -\frac{1}{a(n-1)\sin^{n-1} ax}+\lnint\left|frac{dx}{(\tansin^{n-2} ax)(\right|+Ccos ax)} \qquad\mbox{(}n\neq 1\mbox{)}</math>
 
: <math>\int\frac{\sin ax\,dx}{\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax} +C\qquad\mbox{(}n\neq 1\mbox{)}</math>
:
 
: <math>\int\frac{dx}{\sin^n2 ax\,dx}{\cos ax} = -\frac{1}{a(n-1)}\sin^ ax+\frac{n-1} ax{a}+\intln\left|\tan\left(\frac{dx\pi}{4}+\sin^frac{ax}{n-2} ax\cos ax} \qquad(n\neq 1right)\,\!right|+C</math>
<math>\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} \qquad(n\neq 1)\,\!</math>
: <math>\int\frac{\sin ax\;dx}{\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax} +C\qquad(n\neq 1)\,\!</math>
 
: <math>\int\frac{\sin^2 ax\;,dx}{\cos^n ax} = -\frac{1\sin ax}{a}(n-1)\sin cos^{n-1}ax+}-\frac{1}{an-1}\ln\left|\tan\left(int\frac{\pidx}{4}+\fraccos^{n-2}ax} \qquad\mbox{2(}n\right)neq 1\right|+Cmbox{)}</math>
 
: <math>\int\frac{\sin^2n ax\;,dx}{\cos^n ax} = -\frac{\sin ax}{a(n-1)\cos^{n-1} ax}-\frac{1}{a(n-1)} + \int\frac{dx}{\cossin^{n-2} ax\,dx}{\cos ax} \qquad\mbox{(}n\neq 1)\,\!mbox{)}</math>
 
: <math>\int\frac{\sin^n ax\;,dx}{\cos^m ax} = -\fracbegin{\sin^{n-1cases} ax}{a(n-1)} + \int\frac{\sin^{n-2} ax\;dx}{\cos ax} \qquad(n\neq 1)\,\!</math>
\dfrac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\dfrac{n-m+2}{m-1}\displaystyle\int\dfrac{\sin^n ax\,dx}{\cos^{m-2} ax} &\mbox{(}m\neq 1\mbox{)} \\
\dfrac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\dfrac{n-1}{m-1}\displaystyle\int\dfrac{\sin^{n-2} ax\,dx}{\cos^{m-2} ax} &\mbox{(}m\neq 1\mbox{)} \\
-\dfrac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\dfrac{n-1}{n-m}\displaystyle\int\dfrac{\sin^{n-2} ax\,dx}{\cos^m ax} &\mbox{(}m\neq n\mbox{)}
\end{cases}</math>
 
: <math>\int\frac{\sin^ncos ax\;,dx}{\cossin^mn ax} = -\frac{\sin^{n+1} ax}{a(mn-1)\cossin^{mn-1} ax}-\frac{n-m +2}{m-1}C\intqquad\fracmbox{\sin^(}n ax\;dx}{\cos^{m-2} ax} \qquad(m\neq 1\mbox{)\,\!}</math>
: và: <math>\int\frac{\sin^n ax\;dx}{\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} ax\;dx}{\cos^m ax} \qquad(m\neq n)\,\!</math>
: và: <math>\int\frac{\sin^n ax\;dx}{\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} ax\;dx}{\cos^{m-2} ax} \qquad(m\neq 1)\,\!</math>
 
: <math>\int\frac{\cos^2 ax\;,dx}{\sin^n ax} = -\frac{1}{a}\left(n-1)\sin^{n-1}cos ax} +C\qquad(nln\neq 1)left|\,tan\!frac{ax}{2}\right|\right) +C</math>
 
: <math>\int\frac{\cos^2 ax\;,dx}{\sin^n ax} = -\frac{1}{an-1}\left(\frac{\cos ax}{a\sin^{n-1} ax}+\ln\left|\tanint\frac{axdx}{\sin^{n-2}\right| ax}\right) +C\qquad\mbox{(}n\neq 1\mbox{)}</math>
 
: <math>\int\frac{\cos^2n ax\;,dx}{\sin^nm ax} = -\fracbegin{1cases}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax)}+\int\frac{dx}{\sin^{n-2} ax}\right) \qquad(n\neq 1)</math>
-\dfrac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \dfrac{n-m+2}{m-1}\displaystyle\int\dfrac{\cos^n ax\,dx}{\sin^{m-2} ax} &\mbox{(}m\neq 1\mbox{)} \\
 
: <math>\int\frac{\cos^n ax\;dx}{\sin^m ax} = -\fracdfrac{\cos^{n+-1} ax}{a(m-1)\sin^{m-1} ax} - \fracdfrac{n-m-21}{m-1}\displaystyle\int\fracdfrac{\cos^{n-2} ax\;,dx}{\sin^{m-2} ax} &\qquadmbox{(}m\neq 1\mbox{)} \,\!</math>
: và: <math>\int\frac{\cos^n ax\;dx}{\sin^m ax} = \fracdfrac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \fracdfrac{n-1}{n-m}\displaystyle\int\fracdfrac{\cos^{n-2} ax\;,dx}{\sin^m ax} &\qquadmbox{(}m\neq n)\,\!</math>mbox{)}
\end{cases}</math>
: và: <math>\int\frac{\cos^n ax\;dx}{\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} ax\;dx}{\sin^{m-2} ax} \qquad(m\neq 1)\,\!</math>
 
== Tích phân chứa hàm [[sin]] và [[hàm lượng giác|tang]] ==
 
: <math>\int \sin ax \tan ax\;,dx = \frac{1}{a}(\ln|\sec ax + \tan ax| - \sin ax)+C\,\!</math>
 
: <math>\int\frac{\tan^n ax\;dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) +C\qquad(n\neq 1)\,\!</math>
: <math>\int\frac{\tan^n ax\,dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) +C\qquad(n\neq 1)\,\!</math>
 
== Tích phân chứa hàm [[hàm lượng giác|cos]] và [[hàm lượng giác|tang]] ==
 
: <math>\int\frac{\tan^n ax\;,dx}{\cos^2 ax} = \frac{1}{a(n+1)}\tan^{n+1} ax +C\qquad(n\neq -1)\,\!</math>
 
== Tích phân chứa hàm [[sin]] và [[hàm lượng giác|cotang]] ==
 
: <math>\int\frac{\cot^n ax\;,dx}{\sin^2 ax} = -\frac{1}{a(n+1)}\cot^{n+1} ax +C\qquad(n\neq -1)\,\!</math>
 
== Tích phân chứa hàm [[hàm lượng giác|cos]] và [[hàm lượng giác|cotang]] ==
 
: <math display="block">\int\frac{\cot^n ax\;,dx}{\cos^2 ax} = \frac{1}{a(1-n)}\tan^{1-n} ax +C\qquad(n\neq 1)\,\!</math>
 
== Tích phân chứa hàm [[Hàm lượng giác|secant]] và [[Hàm lượng giác|tang]] ==
 
: <math> \int(\sec x)(\tan x)\,dx= \sec x + C</math>
 
== Tích phân chứa hàm [[Hàm lượng giác|cosecant]] và [[cotang]] ==
 
: <math> \int(\csc x)(\cot x)\,dx= -\csc x + C</math>
 
== Tích phân trên một phần tư đường tròn ==
: <math>\int_{{0}}^{{\frac{\pi}{2}}} \sin^n x \, dx = \int_{{0}}^{{\frac{\pi}{2}}} \cos^n x \, dx = \begin{cases}
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n=2,4,6,8,\ldots \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, & n=3,5,7,9,\ldots \\
1, & n=1
\end{cases}</math>
 
== Tích phân với giới hạn đối xứng ==
 
: <math>\int_{{-c}}^{{c}}\sin {x}\;,dx = 0 \!</math>
 
: <math>\int_{{-c}}^{{c}}\cos {x}\;dx = 2\int_{{0}}^{{c}}\cos {x}\;dx = 2\int_{{-c}}^{{0}}\cos {x}\;dx = 2\sin {c} \!</math>
: <math>\int_{{-c}}^{{c}}\tancos {x}\;,dx = 2\int_{{0}}^{{c}}\cos {x}\!,dx = 2\int_{{-c}}^{{0}}\cos {x}\,dx = 2\sin {c} </math>
 
: <math>\int_{-\frac{a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\;dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad(n=1,3,5...)\,\!</math>
: <math>\int_{{-c}}^{{c}}\tan {x}\,dx = 0 </math>
 
: <math>\int_{-\frac{a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\,dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad</math> ({{mvar|n}} là số nguyên dương lẻ)
 
: <math>\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\,dx = \frac{a^3(n^2\pi^2-6(-1)^n)}{24n^2\pi^2} = \frac{a^3}{24} (1-6\frac{(-1)^n}{n^2\pi^2}) \qquad</math> ({{mvar|n}} là số nguyên dương)
 
== Tích phân trên toàn bộ đường tròn ==
 
: <math>\int_{{0}}^{{2 \pi}}\sin^{2m+1}{x}\cos^{2n+1}{x}\,dx = 0 \qquad m,n \in \mathbb{Z}</math>
 
==Tham khảo==
{{Tham khảo}}
* {{cite book | last=Gradshteĭn | first=I. S. | title=Table of Integrals, Series, and Products | publisher=Academic Press | publication-place=Waltham, MA | year=2015 | isbn=978-0-12-384933-5 | oclc=893676501}}
 
{{Danh sách tích phân}}
 
[[Thể loại:Tích phân|Hàm lượng giác]]