Список интегралов от тригонометрических функций

Ниже приведён список интегралов (первообразных функций) от тригонометрических функций. В списке везде опущена аддитивная константа интегрирования.

Константа ${\displaystyle c}$ не равняется нулю.

${\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx+C}$

${\displaystyle \int \sin ^{n}cx\;dx=-{\frac {\sin ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \sin ^{n-2}cx\;dx\qquad {\mbox{( }}n>0{\mbox{)}}}$

${\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}}$

${\displaystyle \int x^{2}\sin cx\;dx={\frac {2\cos cx}{c^{3}}}+{\frac {2x\sin cx}{c^{2}}}-{\frac {x^{2}\cos cx}{c}}}$

${\displaystyle \int x^{3}\sin cx\;dx=-{\frac {6\sin cx}{c^{4}}}+{\frac {6x\cos cx}{c^{3}}}+{\frac {3x^{2}\sin cx}{c^{2}}}-{\frac {x^{3}\cos cx}{c}}}$

${\displaystyle \int x^{4}\sin cx\;dx=-{\frac {24\cos cx}{c^{5}}}-{\frac {24x\sin cx}{c^{4}}}+{\frac {12x^{2}\cos cx}{c^{3}}}+{\frac {4x^{3}\sin cx}{c^{2}}}-{\frac {x^{4}\cos cx}{c}}}$

${\displaystyle \int x^{5}\sin cx\;dx={\frac {120\sin cx}{c^{6}}}-{\frac {120x\cos cx}{c^{5}}}-{\frac {60x^{2}\sin cx}{c^{4}}}+{\frac {20x^{3}\cos cx}{c^{3}}}+{\frac {5x^{4}\sin cx}{c^{2}}}-{\frac {x^{5}\cos cx}{c}}}$

${\displaystyle {\begin{aligned}\int x^{n}\sin cx\;dx&=n!\cdot \sin cx\left[{\frac {x^{n-1}}{c^{2}\cdot (n-1)!}}-{\frac {x^{n-3}}{c^{4}\cdot (n-3)!}}+{\frac {x^{n-5}}{c^{6}\cdot (n-5)!}}-...\right]-\\&-n!\cdot \cos cx\left[{\frac {x^{n}}{c\cdot n!}}-{\frac {x^{n-2}}{c^{3}\cdot (n-2)!}}+{\frac {x^{n-4}}{c^{5}\cdot (n-4)!}}-...\right]\end{aligned}}}$

${\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{( }}n\geq 0{\mbox{)}}}$

${\displaystyle \int {\frac {\sin cx}{x}}dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}}$

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

${\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}$

${\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{( }}n>1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\operatorname {tg} \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}$

${\displaystyle \int {\frac {x\;dx}{1+\sin cx}}={\frac {x}{c}}\operatorname {tg} \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}$

${\displaystyle \int {\frac {x\;dx}{1-\sin cx}}={\frac {x}{c}}\operatorname {ctg} \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}$

${\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\operatorname {tg} \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}$

${\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin((c_{1}-c_{2})x)}{2(c_{1}-c_{2})}}-{\frac {\sin((c_{1}+c_{2})x)}{2(c_{1}+c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}}$

${\displaystyle \int \cos cx\;dx={\frac {1}{c}}\sin cx+C}$

${\displaystyle \int \cos ^{n}cx\;dx={\frac {\cos ^{n-1}cx\sin cx}{nc}}+{\frac {n-1}{n}}\int \cos ^{n-2}cx\;dx\qquad {\mbox{( }}n>0{\mbox{)}}}$

${\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}}}+{\frac {x\sin cx}{c}}}$

${\displaystyle \int x^{n}\cos cx\;dx={\frac {x^{n}\sin cx}{c}}-{\frac {n}{c}}\int x^{n-1}\sin cx\;dx}$

${\displaystyle \int {\frac {\cos cx}{x}}dx=\ln |cx|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}}{2i\cdot (2i)!}}}$

${\displaystyle \int {\frac {\cos cx}{x^{n}}}dx=-{\frac {\cos cx}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\sin cx}{x^{n-1}}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\cos cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}$

${\displaystyle \int {\frac {dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{( }}n>1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{1+\cos cx}}={\frac {1}{c}}\operatorname {tg} {\frac {cx}{2}}}$

${\displaystyle \int {\frac {dx}{1-\cos cx}}=-{\frac {1}{c}}\operatorname {ctg} {\frac {cx}{2}}}$

${\displaystyle \int {\frac {x\;dx}{1+\cos cx}}={\frac {x}{c}}\operatorname {tg} {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\cos {\frac {cx}{2}}\right|}$

${\displaystyle \int {\frac {x\;dx}{1-\cos cx}}=-{\frac {x}{c}}\operatorname {ctg} {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\sin {\frac {cx}{2}}\right|}$

${\displaystyle \int {\frac {\cos cx\;dx}{1+\cos cx}}=x-{\frac {1}{c}}\operatorname {tg} {\frac {cx}{2}}}$

${\displaystyle \int {\frac {\cos cx\;dx}{1-\cos cx}}=-x-{\frac {1}{c}}\operatorname {ctg} {\frac {cx}{2}}}$

${\displaystyle \int \cos c_{1}x\cos c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}}$

${\displaystyle \int \operatorname {tg} cx\;dx=-{\frac {1}{c}}\ln |\cos cx|}$

${\displaystyle \int \operatorname {tg} ^{n}cx\;dx={\frac {1}{c(n-1)}}\operatorname {tg} ^{n-1}cx-\int \operatorname {tg} ^{n-2}cx\;dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|}$

${\displaystyle \int {\frac {dx}{\operatorname {tg} cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}$

${\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|}$

${\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}$

${\displaystyle \int \sec {cx}\,dx={\frac {1}{c}}\ln {\left|\sec {cx}+\operatorname {tg} {cx}\right|}}$

${\displaystyle \int \sec ^{n}{cx}\,dx={\frac {\sec ^{n-1}{cx}\sin {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{cx}\,dx\qquad {\mbox{ ( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\operatorname {tg} {\frac {x}{2}}}$

${\displaystyle \int \operatorname {cosec} {cx}\,dx=-{\frac {1}{c}}\ln {\left|\operatorname {cosec} {cx}+\operatorname {ctg} {cx}\right|}}$

${\displaystyle \int \operatorname {cosec} ^{n}{cx}\,dx=-{\frac {\operatorname {cosec} ^{n-1}{cx}\cos {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \operatorname {cosec} ^{n-2}{cx}\,dx\qquad {\mbox{ ( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \operatorname {ctg} cx\;dx={\frac {1}{c}}\ln |\sin cx|}$

${\displaystyle \int \operatorname {ctg} ^{n}cx\;dx=-{\frac {1}{c(n-1)}}\operatorname {ctg} ^{n-1}cx-\int \operatorname {ctg} ^{n-2}cx\;dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{1+\operatorname {ctg} cx}}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}}}$

${\displaystyle \int {\frac {dx}{1-\operatorname {ctg} cx}}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}}}$

${\displaystyle \int {\frac {dx}{\cos cx\pm \sin cx}}={\frac {1}{c{\sqrt {2}}}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|}$

${\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\operatorname {tg} \left(cx\mp {\frac {\pi }{4}}\right)}$

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

${\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}$

${\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}$

${\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+\cos cx)}}=-{\frac {1}{4c}}\operatorname {tg} ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}$

${\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1-\cos cx)}}=-{\frac {1}{4c}}\operatorname {ctg} ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1+\sin cx)}}={\frac {1}{4c}}\operatorname {ctg} ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1-\sin cx)}}={\frac {1}{4c}}\operatorname {tg} ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\operatorname {tg} \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}$

${\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}}\sin ^{2}cx}$

${\displaystyle \int \sin c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}\qquad {\mbox{( }}|c_{1}|\neq |c_{2}|{\mbox{)}}}$

${\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx=-{\frac {\sin ^{n-1}cx\cos ^{m+1}cx}{c(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}cx\cos ^{m}cx\;dx\qquad {\mbox{( }}m,n>0{\mbox{)}}}$

${\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx={\frac {\sin ^{n+1}cx\cos ^{m-1}cx}{c(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}cx\cos ^{m-2}cx\;dx\qquad {\mbox{( }}m,n>0{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} cx\right|}$

${\displaystyle \int {\frac {dx}{\sin cx\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}+\int {\frac {dx}{\sin cx\cos ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\sin ^{n}cx\cos cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}+\int {\frac {dx}{\sin ^{n-2}cx\cos cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos cx}}=-{\frac {1}{c}}\sin cx+{\frac {1}{c}}\ln \left|\operatorname {tg} \left({\frac {\pi }{4}}+{\frac {cx}{2}}\right)\right|}$

${\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos cx}}=-{\frac {\sin ^{n-1}cx}{c(n-1)}}+\int {\frac {\sin ^{n-2}cx\;dx}{\cos cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n+1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}=-{\frac {\sin ^{n-1}cx}{c(n-m)\cos ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}cx\;dx}{\cos ^{m}cx}}\qquad {\mbox{( }}m\neq n{\mbox{)}}}$

${\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-1}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|\right)}$

${\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{n-1}}\left({\frac {\cos cx}{c\sin ^{n-1}cx)}}+\int {\frac {dx}{\sin ^{n-2}cx}}\right)\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n+1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-m-2}{m-1}}\int {\frac {cos^{n}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}={\frac {\cos ^{n-1}cx}{c(n-m)\sin ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m}cx}}\qquad {\mbox{( }}m\neq n{\mbox{)}}}$

${\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n-1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{( }}m\neq 1{\mbox{)}}}$

${\displaystyle \int \sin cx\operatorname {tg} cx\;dx={\frac {1}{c}}(\ln |\sec cx+\operatorname {tg} cx|-\sin cx)}$

${\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n-1)}}\operatorname {tg} ^{n-1}(cx)\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\operatorname {tg} ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\operatorname {tg} ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}}$

${\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n+1)}}\operatorname {ctg} ^{n+1}cx\qquad {\mbox{( }}n\neq -1{\mbox{)}}}$

${\displaystyle \int {\frac {\operatorname {ctg} ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\operatorname {tg} ^{1-n}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\operatorname {tg} ^{m}(cx)}{\operatorname {ctg} ^{n}(cx)}}\;dx={\frac {1}{c(m+n-1)}}\operatorname {tg} ^{m+n-1}(cx)-\int {\frac {\operatorname {tg} ^{m-2}(cx)}{\operatorname {ctg} ^{n}(cx)}}\;dx\qquad {\mbox{( }}m+n\neq 1{\mbox{)}}}$

Книги

• Градштейн И. С., Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений (рус.). — 4-е изд. — М.: Наука, 1963.
• Двайт Г. Б. Таблицы интегралов (рус.). — СПб.: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
• Zwillinger D. CRC Standard Mathematical Tables and Formulae (англ.). — 31st ed. — 2002. — ISBN 1-58488-291-3.
• Справочник по специальным функциям с формулами, графиками и таблицами / Под ред. М. Абрамовица и И. Стиган; пер. с англ. под ред. В. А. Диткина и Л. Н. Карамзиной. — М.: Наука, 1979. — 832 с. — 50 000 экз.
• Корн Г. А., Корн Т. М. Справочник по математике для научных работников и инженеров. — М.: «Наука», 1974. — 832 с.

Таблицы интегралов

• Интегралы на EqWorld
• S.O.S. Mathematics: Tables and Formulas

Вычисление интегралов

• The Integrator (на Wolfram Research)
• Империя Чисел
• Методы вычисления неопределённых интегралов