Վիքիպեդիայից՝ ազատ հանրագիտարանից
Եռանկյունաչափական ֆունկցիաների ինտեգրալների ցանկ, ստորև ներկայացված են սինուս, կոսինուս, տանգենս, կոտանգենս, սեկանս, կոսեկանս ֆունկցիաների ինտեգրալների ցանկերը։ Անորոշ ինտեգրալների համար ինտեգրման հաստատունը բաց է թողնված։
![{\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90927a9524ca2ebb7532b3b642442416f997074c)
![{\displaystyle \int \sin ^{n}cx\;dx=-{\frac {\sin ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \sin ^{n-2}cx\;dx\qquad {\mbox{( }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30672cff02526fa53ba8cad14a342f4e7f5e9a4)
![{\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba196bb420de05f3a536e722b9fa9b633ae1edb)
![{\displaystyle \int x^{2}\sin cx\;dx={\frac {2\cos cx}{c^{3}}}+{\frac {2x\sin cx}{c^{2}}}-{\frac {x^{2}\cos cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4061e9a8488ee96076f947cc77a53baf236af218)
![{\displaystyle \int x^{3}\sin cx\;dx=-{\frac {6\sin cx}{c^{4}}}+{\frac {6x\cos cx}{c^{3}}}+{\frac {3x^{2}\sin cx}{c^{2}}}-{\frac {x^{3}\cos cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a02dbba2bf776aeb40b8258db9fa4d464b5ec57)
![{\displaystyle \int x^{4}\sin cx\;dx=-{\frac {24\cos cx}{c^{5}}}-{\frac {24x\sin cx}{c^{4}}}+{\frac {12x^{2}\cos cx}{c^{3}}}+{\frac {4x^{3}\sin cx}{c^{2}}}-{\frac {x^{4}\cos cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8545511869803d49269711116eda54f540e6c779)
![{\displaystyle \int x^{5}\sin cx\;dx={\frac {120\sin cx}{c^{6}}}-{\frac {120x\cos cx}{c^{5}}}-{\frac {60x^{2}\sin cx}{c^{4}}}+{\frac {20x^{3}\cos cx}{c^{3}}}+{\frac {5x^{4}\sin cx}{c^{2}}}-{\frac {x^{5}\cos cx}{c}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f7bcd22c2d6f86054a6f2f747773a9d7cda1f1)
![{\displaystyle {\begin{aligned}\int x^{n}\sin cx\;dx&=n!\cdot \sin cx\left[{\frac {x^{n-1}}{c^{2}\cdot (n-1)!}}-{\frac {x^{n-3}}{c^{4}\cdot (n-3)!}}+{\frac {x^{n-5}}{c^{6}\cdot (n-5)!}}-...\right]-\\&-n!\cdot \cos cx\left[{\frac {x^{n}}{c\cdot n!}}-{\frac {x^{n-2}}{c^{3}\cdot (n-2)!}}+{\frac {x^{n-4}}{c^{5}\cdot (n-4)!}}-...\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7386e0080c6581e6c60f98d21aa1ea15e010715d)
![{\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{( }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85fd8899dc119d4fd51b3d062b5b611cae4b2765)
![{\displaystyle \int {\frac {\sin cx}{x}}dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c679a50c11f75c5624254025badb770884691f)
![{\displaystyle \int {\frac {\sin cx}{x^{n}}}dx=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2c4bcf48fc3ff6ee0d8eb14b330bf55dacd67f)
![{\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ebe505eddb09fdee5a96ed81dd932c9aebaab95)
![{\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{( }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b9283e5fb23994b2a182abad3af4409f0954ce)
![{\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\operatorname {tg} \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2260b4cd2622705ef4afc009592c6bf59afa59)
![{\displaystyle \int {\frac {x\;dx}{1+\sin cx}}={\frac {x}{c}}\operatorname {tg} \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1e52675332df6b4c1108fc3ced9f4cea50f3b8)
![{\displaystyle \int {\frac {x\;dx}{1-\sin cx}}={\frac {x}{c}}\operatorname {ctg} \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2145f2d63bf34df76e5181ad30d49b7ad4b5a76)
![{\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\operatorname {tg} \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c656cce2f8bc238e2d2204b985648562451540)
![{\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6951ac819e94bc652e8fd1bf5ae804bd64efee0e)
![{\displaystyle \int \cos ax\;\mathrm {d} x={\frac {1}{a}}\sin ax+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf48148c2f2551f320d14b44087faade82eff3b8)
![{\displaystyle \int \cos ^{2}{ax}\;\mathrm {d} x={\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/22f3a549538075b161710d3de8ed8530a53e31d7)
![{\displaystyle \int \cos ^{n}ax\;\mathrm {d} x={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed015aabd8d89a8728a771e9fa19df9b4e941b0)
![{\displaystyle \int x\cos ax\;\mathrm {d} x={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de4f46e55c023e737c26af3d41639140d31f585f)
![{\displaystyle \int x^{2}\cos ^{2}{ax}\;\mathrm {d} x={\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/77e9a51f8c9a107ed7d2c954db63e8c8102a8dea)
![{\displaystyle \int x^{n}\cos ax\;\mathrm {d} x={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;\mathrm {d} x\,=\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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f83f28bfabedbfc669233383f1cdf5348c759fbb)
![{\displaystyle \int {\frac {\cos ax}{x}}\mathrm {d} x=\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/ef9927ebeaec09fab122d7891bafa8ef9aaff66f)
![{\displaystyle \int {\frac {\cos ax}{x^{n}}}\mathrm {d} x=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a84c4e0dcb38f68d37353a30c20da666dbb795bf)
![{\displaystyle \int {\frac {\mathrm {d} x}{\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/103d2bcc98cdf888bd853f8f794b825c0c6b3ee5)
![{\displaystyle \int {\frac {\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3691fdd78f99348b2dad004ff75b489eefeaeb3)
![{\displaystyle \int {\frac {\mathrm {d} x}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2db02bf89ec890b6e841f68304d7e28a3edde51f)
![{\displaystyle \int {\frac {\mathrm {d} x}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25c98b3e02e04094379615e1533038142f86e1f4)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{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/affe021010cfe4ed02f1b174837899768ac93cca)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{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/a20539258a72e739c7c6b7233c123087064cc610)
![{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeba7fad7f702e58fd2b72d5a0a8f71e2dd62c86)
![{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edfa05cb70dff2af793d724e554871a862f28b89)
![{\displaystyle \int \cos a_{1}x\cos a_{2}x\;\mathrm {d} x={\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{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e16f64b59fabd69040d3106eb7c090d0c5656c5c)
![{\displaystyle \int \operatorname {tg} cx\;dx=-{\frac {1}{c}}\ln |\cos cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd013be01f93e7db05eacdd825ecda1bf32cc9e)
![{\displaystyle \int \operatorname {tg} ^{n}cx\;dx={\frac {1}{c(n-1)}}\operatorname {tg} ^{n-1}cx-\int \operatorname {tg} ^{n-2}cx\;dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e503a55333c41b7533b037befcfd5cc4e1e4744)
![{\displaystyle \int {\frac {dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9bcfd07e2f7bd3dff8ea917d3e242f2ed6f8dc9)
![{\displaystyle \int {\frac {dx}{\operatorname {tg} cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff57a6b757cbe2923ff9e203a7158a69c28d454)
![{\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24aba584cb406b56d201fe031c5ae0fb8b897099)
![{\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c6c283ceb0caa9ef3bbdfd6a6c5fcf271fa29fc)
![{\displaystyle \int \cot ax\;\mathrm {d} x={\frac {1}{a}}\ln |\sin ax|+C\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54449d79cb807b69a1d7e1dc0692fb2aea914199)
![{\displaystyle \int \cot ^{n}ax\;\mathrm {d} x=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b44972559d7106e2a84220c61cdb9d5bc1f1207b)
![{\displaystyle \int {\frac {\mathrm {d} x}{1+\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2b19a3daee6179aaa42651542fae13c161555c)
![{\displaystyle \int {\frac {\mathrm {d} x}{1-\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948489d6f3978ce2b0bb416f93716260f64a04ee)
![{\displaystyle \int \sec {ax}\,\mathrm {d} x={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec334e9560b0664b9c86445ef967255f6104c398)
![{\displaystyle \int \sec ^{2}{x}\,\mathrm {d} x=\tan {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca63419cb8d63f84553f904b221eecbca813037f)
![{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c93596f1fc36e72157fb7ba557148a5be5b26a65)
![{\displaystyle \int \sec ^{n}{ax}\,\mathrm {d} x={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62dc65596e4c2444f35c6ef2932e1914bfced296)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bf13c8fd9c6fe5b56e33503d3c1aaeac4f3040)
![{\displaystyle \int \csc {cx}\,dx=-{\frac {1}{c}}\ln {\left|\csc {cx}+\operatorname {ctg} {cx}\right|}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a98269757127360a0e99a7406a493ad5ce5308)
![{\displaystyle \int \csc ^{n}{cx}\,dx=-{\frac {\csc ^{n-1}{cx}\cos {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{cx}\,dx\qquad {\mbox{ ( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bf9348ae0aa7facb3c7cad62890069c5b74a8e)
Ինտեգրալներ խառը եռանկյունաչափական ֆունկցիաներով[խմբագրել | խմբագրել կոդը]
![{\displaystyle \int {\frac {dx}{\cos cx\pm \sin cx}}={\frac {1}{c{\sqrt {2}}}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fce78ede783c3d6d4292e3c04e3d5513d138f742)
![{\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\operatorname {tg} \left(cx\mp {\frac {\pi }{4}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f81b788821fc0362cce952b6eff3f5affb0b03eb)
![{\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 cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7c28a3cbb8faae8da164304c18ffac19bd99cd)
![{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641492bf3f2102ff0e0a3c3aab3fadb29fd74c4e)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/948e3466f7b59d4a7e6e34fa5c4a5f6536ad7420)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/635f3cb11427290dd7b1146eea072bd34274c244)
![{\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+\cos cx)}}=-{\frac {1}{4c}}\operatorname {tg} ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03119628653a3269c34414d02c353604a2a21d34)
![{\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+-\cos cx)}}=-{\frac {1}{4c}}\operatorname {ctg} ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44135dac5864820fcbbb430d17a02f5182385fcd)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1+\sin cx)}}={\frac {1}{4c}}\operatorname {ctg} ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11a5bd324f6bc37d4a4046855e94a727321ceded)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1-\sin cx)}}={\frac {1}{4c}}\operatorname {tg} ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e958e0dc372fde93e01865bd1b3937ddd02260e7)
![{\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}}\sin ^{2}cx\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b23856c5dc1d08f8df89537a8199d6cc6e631b2)
![{\displaystyle \int \sin c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fcf57b17e9d46e74d55a94319b42e07e3b3d1d7)
![{\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82ed43753482d7f4f9203c866999bd03417494a0)
![{\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8364b79ca100f1c2495b2d0498f3d337ee6bc83)
![{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx=-{\frac {\sin ^{n-1}cx\cos ^{m+1}cx}{c(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}cx\cos ^{m}cx\;dx\qquad {\mbox{( }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88a26cd522e2f07da8ea2e479dcaa5223297c876)
![{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx={\frac {\sin ^{n+1}cx\cos ^{m-1}cx}{c(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}cx\cos ^{m-2}cx\;dx\qquad {\mbox{( }}m,n>0{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a28de0422c56b03f9dede1f1818630db1bb2ef92)
![{\displaystyle \int {\frac {dx}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} cx\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cad2d3cefa651ae06dcce789c88bc0ad4f54cf6)
![{\displaystyle \int {\frac {dx}{\sin cx\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}+\int {\frac {dx}{\sin cx\cos ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa55e16bb7ae2475fe29834529727df4413014f3)
![{\displaystyle \int {\frac {dx}{\sin ^{n}cx\cos cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}+\int {\frac {dx}{\sin ^{n-2}cx\cos cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/737434945c66cd0d4b146d07de294a71dd3364b1)
![{\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/218083942f7cd58e7dc1997647c62595fb1ec9e2)
![{\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos cx}}=-{\frac {1}{c}}\sin cx+{\frac {1}{c}}\ln \left|\operatorname {tg} \left({\frac {\pi }{4}}+{\frac {cx}{2}}\right)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c829b577e6c22b48be7c4dfb9e1760e2ea9632)
![{\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c6dad4787a8dc281f8451ebe694df2ccf40418)
![{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos cx}}=-{\frac {\sin ^{n-1}cx}{c(n-1)}}+\int {\frac {\sin ^{n-2}cx\;dx}{\cos cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8695d54f1af4201bbe882ad2303c3145ead92056)
![{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n+1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b1c5f14597750be6b90ee7b919c57485ed56ca)
![{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}=-{\frac {\sin ^{n-1}cx}{c(n-m)\cos ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}cx\;dx}{\cos ^{m}cx}}\qquad {\mbox{( }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed219a4f688c0586cdb990abfd3df77a8540a38b)
![{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-1}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bfec92405b5f9438bf1538079169cd0d877cb0)
![{\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a32a5742c1035350af73ac78e10e7f25760f2e8)
![{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/049041b9d8157d31b7201fef4c89ec4341063d7d)
![{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{n-1}}\left({\frac {\cos cx}{c\sin ^{n-1}cx)}}+\int {\frac {dx}{\sin ^{n-2}cx}}\right)\qquad {\mbox{( }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3124c5956f6c9de145770eedd1533f4bad845123)
![{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n+1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-m-2}{m-1}}\int {\frac {cos^{n}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eca6226eeb75a120832c81cd8d4b3ef08678f0d)
![{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}={\frac {\cos ^{n-1}cx}{c(n-m)\sin ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m}cx}}\qquad {\mbox{( }}m\neq n{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/264a025ca07841c7ec58f6d9015726bf5eda2b12)
![{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n-1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5687ad6dd6bcce37b9605c98a53580687e1f45)
![{\displaystyle \int \sin cx\operatorname {tg} cx\;dx={\frac {1}{c}}(\ln |\sec cx+\operatorname {tg} cx|-\sin cx)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed61fbb1910ab0bb38306408425f0f5f8c242197)
![{\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n-1)}}\operatorname {tg} ^{n-1}(cx)\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9aff72766792830d38d5fa09b12fde0b9cbbfa)
![{\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\operatorname {tg} ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e726c25c09da96ad244d8e9cc1c867858bfdf2)
![{\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n+1)}}\operatorname {ctg} ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad415958aa72a94a36472193599dda2b438c4b3f)
![{\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\operatorname {tg} ^{1-n}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bce40e9ff6fca226b7e89b923d961f44cd31f688)
![{\displaystyle \int {\frac {\operatorname {tg} ^{m}(cx)}{\operatorname {ctg} ^{n}(cx)}}\;dx={\frac {1}{c(m+n-1)}}\operatorname {tg} ^{m+n-1}(cx)-\int {\frac {\operatorname {tg} ^{m-2}(cx)}{\operatorname {ctg} ^{n}(cx)}}\;dx\qquad {\mbox{( }}m+n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6b8e8bda8ceac9aadf509ff2697b8e55bda0bc)
- Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений (4-е издание). М.։ Наука, 1963. ISBN 0-12-294757-6 // EqWorld
- Двайт Г. Б. Таблицы интегралов СПб։ «Издательство и типография АО ВНИИГ им. Б. В. Веденеева», 1995.-176 с. ISBN 5-85529-029-8.
- D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
- M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4