Đây là danh sách tích phân (nguyên hàm) của các hàm lượng giác. Đối với tích phân của chứa hàm lượng giác và hàm mũ, xem Danh sách tích phân với hàm mũ. Đối với danh sách đầy đủ các tích phân, xem Danh sách tích phân. Đối với danh sách các tích phân đặc biệt của các hàm lượng giác, xem Tích phân lượng giác.
Nhìn chung, với
là đạo hàm của hàm số
, ta có
![{\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b235833736ee6579828397cf22e6a166efb867be)
Trong mọi công thức dưới đây, a là một hằng số khác không và C ký hiệu cho hằng số tích phân.
![{\displaystyle \int \sin ax\,dx=-{\frac {1}{a}}\cos ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17155f417f1407848abf8090096e58430a91d17a)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94366ec919d67a0f06bb6431029d29b096de77bb)
![{\displaystyle \int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da591fb70e83503367cffa5f8df86754f0181d86)
![{\displaystyle \int x\sin ^{2}{ax}\,dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5aedf47e3b2a17b7f494177d2af59abf5546799)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3902717b8ce182f19cd31aba15f8d9a1cfb72f04)
![{\displaystyle \int x\sin ax\,dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a98562978db18283202b5ee81589cf1c2dc28d4)
![{\displaystyle \int (\sin b_{1}x)(\sin b_{2}x)\,dx={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(}}|b_{1}|\neq |b_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfee1dc451894cedb0222d17c0efa73ec91b4ee)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a302be11097df6cb9bb9d191f5a15743e2a49d06)
![{\displaystyle \int {\frac {dx}{\sin ax}}=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39a4cc7433bffdc5cb3e9a1e92fb0990988bce7d)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35811a5a2048d1a7fac71792e8d4c69c8450a079)
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a78678e73178ca2ed2e6f2d8d46f3e3a4e8aa59)
![{\displaystyle \int {\frac {\sin ax}{x}}\,dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dba87ad697f99f043327f310fe9b7b966fd7943)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1dcfdaf90a90cb3d3e38a3865f92682ab04da8e)
![{\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8439ef42a168ed7e05a7efea83b205790ceb59)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d08dadaff5165e64f02d1efcc3a48ab10c8c8f9e)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2037a630f830c597c4126f076edd449c8967fbea)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1baffb49b75cb47bcc18564e62c50ad40cc37c11)
![{\displaystyle \int \cos ax\,dx={\frac {1}{a}}\sin ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76fe3b3af800a174faece0db14fcdded789dc979)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5a154836333fe3188bc000c0cfe80b86fc8915)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09c4fa076ca236ae144e072ccefc75aa24d248df)
![{\displaystyle \int x\cos ax\,dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d58daa2d9b221f46b811e2a25309b0fcb64c678)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a82a134604ff47f5aecb2a44a092592d160dfc)
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35f3b93b28be99e336a08657ed1029d6ac47bb62)
![{\displaystyle \int {\frac {\cos ax}{x}}\,dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d82b8c7f2081edd47994c4c5600916e3800fb48)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db74e835dd1999aa78221a536dda8d30852fb26d)
![{\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5013bc2428b1006b40c999d6b427a36f5cf0620)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ab71f44d1404eb94b06b1a65a0edfcfeec9d1c)
![{\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41a971cb7f555d9f48a9f2b820bcc7fe53f2436c)
![{\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8d0cb833e9a78d8ea6ff57d1ce08c44aaa09c7)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d39822052ca12c117f7121ef13f59d7fadd8ace)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c71e740a637ad718742a884ab0284c19dcf861)
![{\displaystyle \int {\frac {\cos ax\,dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/868e1340952b82a678a6ca4c964455ffdb51ec09)
![{\displaystyle \int {\frac {\cos ax\,dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/376acd5ba6dda72e5511ca987900ff39f82c3462)
![{\displaystyle \int (\cos a_{1}x)(\cos a_{2}x)\,dx={\frac {\sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})}}+{\frac {\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}}+C\qquad {\mbox{(}}|a_{1}|\neq |a_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e38891f9572c11dbde2be117bccdf8b776b316)
![{\displaystyle \int \tan ax\,dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edf5303e68dccdda4a730769b24a3c96b758afeb)
![{\displaystyle \int \tan ^{2}{x}\,dx=\tan {x}-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83b8d69c24a94eed938f2e751572e874aff74f7f)
![{\displaystyle \int \tan ^{n}ax\,dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\,dx\qquad (n\neq 1)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5278b77a7c4880603c650356f5f6645d3196967c)
![{\displaystyle \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)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0ada76fe7dff158fb4510cc5939e875fb5390e)
![{\displaystyle \int {\frac {dx}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1711694feda807b6b6c4d19c1ccda303ef0c02b)
![{\displaystyle \int {\frac {dx}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acaa4b1fa8ad7c5675d289c229146ea22ebb1ad)
![{\displaystyle \int {\frac {\tan ax\,dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088cf071e50e70f68fc91758aeabaff44a2efc29)
![{\displaystyle \int {\frac {\tan ax\,dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e40e671189477c6db1bb2fef5be206f8e56ad37c)
![{\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0eae695334d259040d728b565ca374a2c89380)
![{\displaystyle \int \sec ^{2}{x}\,dx=\tan {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62448efb9e0512c1014643b2efa34928c397f1b0)
![{\displaystyle \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\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33bd8a9650849a6b65b54d3e27d5e5d3af4fd51e)
[1]
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9acabbd90de19b0d361d572dce3398a57c9d653f)
![{\displaystyle \int {\frac {dx}{\sec {x}-1}}=-x-\cot {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2845a0bb3c940ca6f9d98303dd5944618ad6a93c)
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87d2b18bebeef08259fcc0bd5fec55d0854f8d15)
![{\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/417803af6cef8535c9b9ee74f75a20ab4180fac0)
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/661c6755f2265b575251936f293a8e67dbfe5a71)
![{\displaystyle \int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-2}{ax}\cot {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb3612fab7ac301456c07011e602ef7d5a21b0c)
![{\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2}{\cot {\frac {x}{2}}+1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3116bd8d583f72077e90f06cf8e867997fdd14)
![{\displaystyle \int {\frac {dx}{\csc {x}-1}}=-x+{\frac {2}{\cot {\frac {x}{2}}-1}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12f1dc04dd4f01c7df46f36cba6653112da4418)
![{\displaystyle \int \cot ax\,dx={\frac {1}{a}}\ln |\sin ax|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd8d33638f05fb0f16334bb90a8aa016dc05bca)
![{\displaystyle \int \cot ^{2}{x}\,dx=-\cot {x}-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36979324717e61101b7119e6b3e995d5ec509d69)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59d132e29fd5c6bf2470f0540a37c8234d859be)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f957b8e698c1a8b55ba500a962f1b183b557889e)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e697e83f4089e2905dd245cb432fd219dcc493d)
Tích phân một hàm hữu tỉ (phân thức) của sin và cos có thể được tính bằng quy tắc Bioche.
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99f9f4158d86f68a6f22ac0b494b8df2a009d24)
![{\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e0e3cebd7eac046797eefb5e8be824a6ec6008)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4a31631ace2155c341f3a42e944454f4d2525b)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7b492cf08e982a1acd2fb3a10ed3ac6abb3379)
![{\displaystyle \int {\frac {\sin ax\,dx}{\cos ax\pm \sin ax}}=\pm {\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax\pm \cos ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd94f49bcfe12280445c6a1ea98bf2140a357e8)
![{\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be1a480baee112532376953a0e514953aac55e5)
![{\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5551ba6b1a17809cd93bad200f96d7bd77c41add)
![{\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1+\sin ax)}}={\frac {1}{4a}}\cot ^{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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cba9a5f34c6745abfcdfea7906197167c4bf7fc4)
![{\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1-\sin ax)}}={\frac {1}{4a}}\tan ^{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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89bd9ffed0a439702f128c452dd2e9d363175ef4)
![{\displaystyle \int (\sin ax)(\cos ax)\,dx={\frac {1}{2a}}\sin ^{2}ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd704958c5a62ac35486a94c701c4d4d4d89ae1)
![{\displaystyle \int (\sin a_{1}x)(\cos a_{2}x)\,dx=-{\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\qquad {\mbox{(}}|a_{1}|\neq |a_{2}|{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f83c343fb6df4022ef599ad3cab024ccb4826a1c)
![{\displaystyle \int (\sin ^{n}ax)(\cos ax)\,dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(}}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2264a914e157fe9ae32c0ebcb3560a9c9c6580)
![{\displaystyle \int (\sin ax)(\cos ^{n}ax)\,dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(}}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7ee3b8d8c912480f4ee23b27661e9b6b95b438)
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef7d9f1408bea785c47ea16761dc39c44c0f9e1)
![{\displaystyle \int {\frac {dx}{(\sin ax)(\cos ax)}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03f904edd8835bf1f3b47bce34e30cf3e2fbf32)
![{\displaystyle \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 {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5ecf66b4b0743e67d0a0bf561d9c453b7f10ce)
![{\displaystyle \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)}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/975e45a50d80482ed5dbcce5032dabf4af7e06d6)
![{\displaystyle \int {\frac {\sin ax\,dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3161c999aebad2d3dbb581801f6a9b19da2f5078)
![{\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0901ab7048597a2d942a8d9ec9251ab116891ac8)
![{\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f104fa17fb3e2fbc7bb397bb9a253e1fb96fb341)
![{\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\,dx}{\cos ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4725d1ee07b8bc994a230adba2f1f3d5939c76e1)
![{\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax}}={\begin{cases}{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d0fd4468d9cb4bad93dc052b4cdf7d12f88602)
![{\displaystyle \int {\frac {\cos ax\,dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa430dac06a3f407e75c4dba1895c6e2926f22bf)
![{\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8d5900b25ed3cda6de1b98479c1fc0d1c30cd9)
![{\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax}}+\int {\frac {dx}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fddf45caf47d2a002c3a94ec69e7f7b6436e99c)
![{\displaystyle \int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax}}={\begin{cases}-{\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{)}}\\-{\dfrac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}}-{\dfrac {n-1}{m-1}}\displaystyle \int {\dfrac {\cos ^{n-2}ax\,dx}{\sin ^{m-2}ax}}&{\mbox{(}}m\neq 1{\mbox{)}}\\{\dfrac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}}+{\dfrac {n-1}{n-m}}\displaystyle \int {\dfrac {\cos ^{n-2}ax\,dx}{\sin ^{m}ax}}&{\mbox{(}}m\neq n{\mbox{)}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8d1c5ecd370d56022a4799c9e081f8d1712f6a)
![{\displaystyle \int \sin ax\tan ax\,dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b87e662e0ec75a3c647c72dc0f10bef7f3bd503b)
![{\displaystyle \int {\frac {\tan ^{n}ax\,dx}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad (n\neq 1)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4f2d37e77dac5f6bdb6bc36bf05d5e012a4271)
![{\displaystyle \int {\frac {\tan ^{n}ax\,dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad (n\neq -1)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb64ddb8ea8b9ea11b53883d8c91dd75df5989d)
![{\displaystyle \int {\frac {\cot ^{n}ax\,dx}{\sin ^{2}ax}}=-{\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad (n\neq -1)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/115d95df29301e95799ca0351f1b6585c218c1c4)
![{\displaystyle \int {\frac {\cot ^{n}ax\,dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad (n\neq 1)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61ab90df5fa0a3917a9a8ce89921fdcb86f86320)
![{\displaystyle \int (\sec x)(\tan x)\,dx=\sec x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3426300c895f6ff40c28455d36d29417d683dee)
![{\displaystyle \int (\csc x)(\cot x)\,dx=-\csc x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abd49ac7e4242cab5000f8180c53adcd584240f4)
Tích phân trên một phần tư đường tròn[sửa | sửa mã nguồn]
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7481db21aaa802b14bc2991ee61533439f7abdcf)
Tích phân với giới hạn đối xứng[sửa | sửa mã nguồn]
![{\displaystyle \int _{-c}^{c}\sin {x}\,dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6976aecf1b8b7d692492e777f59de99b7b9b8ac1)
![{\displaystyle \int _{-c}^{c}\cos {x}\,dx=2\int _{0}^{c}\cos {x}\,dx=2\int _{-c}^{0}\cos {x}\,dx=2\sin {c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d17e7b3ec2a19c12316f01d5b4f639bdaee1cce7)
![{\displaystyle \int _{-c}^{c}\tan {x}\,dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e0a8bab0e106ca691271ae26382c544c64c073)
(n là số nguyên dương lẻ)
(n là số nguyên dương)
Tích phân trên toàn bộ đường tròn[sửa | sửa mã nguồn]
![{\displaystyle \int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{2n+1}{x}\,dx=0\qquad m,n\in \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/41a9e5b3ed0e941591a4c642b74c15ff42aa982e)
- ^ Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008