Тригонометрилле функцисен интегралĕсен йышĕ . Аяларах çавнашкал интегралсен (умсăнарсен ) йышне илсе кăтартнă. Пур çĕрте те аддитивлă констаттăна катертнĕ.
Ĕнтĕ тата c {\displaystyle c} кнстанттă нуль мар.
Ĕнтĕ синус çеç пур интегралсем
тӳрлет
∫ tg c x d x = − 1 c ln | cos c x | {\displaystyle \int \operatorname {tg} cx\;dx=-{\frac {1}{c}}\ln |\cos cx|} ∫ tg n c x d x = 1 c ( n − 1 ) tg n − 1 c x − ∫ tg n − 2 c x d x ( n ≠ 1 ) {\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{)}}} ∫ d x tg c x + 1 = x 2 + 1 2 c ln | sin c x + cos c x | {\displaystyle \int {\frac {dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|} ∫ d x tg c x − 1 = − x 2 + 1 2 c ln | sin c x − cos c x | {\displaystyle \int {\frac {dx}{\operatorname {tg} cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|} ∫ tg c x d x tg c x + 1 = x 2 − 1 2 c ln | sin c x + cos c x | {\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|} ∫ tg c x d x tg c x − 1 = x 2 + 1 2 c ln | sin c x − cos c x | {\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|} Ĕнтĕ секанс çеç пур интегралсем
тӳрлет
∫ sec c x d x = 1 c ln | sec c x + tg c x | {\displaystyle \int \sec {cx}\,dx={\frac {1}{c}}\ln {\left|\sec {cx}+\operatorname {tg} {cx}\right|}} ∫ sec n c x d x = sec n − 1 c x sin c x c ( n − 1 ) + n − 2 n − 1 ∫ sec n − 2 c x d x ( n ≠ 1 ) {\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{)}}} ∫ d x sec x + 1 = x − tg x 2 {\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\operatorname {tg} {\frac {x}{2}}}
∫ cosec c x d x = − 1 c ln | cosec c x + ctg c x | {\displaystyle \int \operatorname {cosec} {cx}\,dx=-{\frac {1}{c}}\ln {\left|\operatorname {cosec} {cx}+\operatorname {ctg} {cx}\right|}} ∫ cosec n c x d x = − cosec n − 1 c x cos c x c ( n − 1 ) + n − 2 n − 1 ∫ cosec n − 2 c x d x ( n ≠ 1 ) {\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{)}}}
∫ ctg c x d x = 1 c ln | sin c x | {\displaystyle \int \operatorname {ctg} cx\;dx={\frac {1}{c}}\ln |\sin cx|} ∫ ctg n c x d x = − 1 c ( n − 1 ) ctg n − 1 c x − ∫ ctg n − 2 c x d x ( n ≠ 1 ) {\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{)}}} ∫ d x 1 + ctg c x = ∫ tg c x d x tg c x + 1 {\displaystyle \int {\frac {dx}{1+\operatorname {ctg} cx}}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}}} ∫ d x 1 − ctg c x = ∫ tg c x d x tg c x − 1 {\displaystyle \int {\frac {dx}{1-\operatorname {ctg} cx}}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}}} Ĕнтĕ синус тата косинус çеç пур интегралсем
тӳрлет
∫ d x cos c x ± sin c x = 1 c 2 ln | tg ( c x 2 ± π 8 ) | {\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|} ∫ d x ( cos c x ± sin c x ) 2 = 1 2 c tg ( c x ∓ π 4 ) {\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\operatorname {tg} \left(cx\mp {\frac {\pi }{4}}\right)} ∫ d x ( cos x + sin x ) n = 1 n − 1 ( sin x − cos x ( cos x + sin x ) n − 1 − 2 ( n − 2 ) ∫ d x ( cos x + sin x ) n − 2 ) {\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)} ∫ cos c x d x cos c x + sin c x = x 2 + 1 2 c ln | sin c x + cos c x | {\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|} ∫ cos c x d x cos c x − sin c x = x 2 − 1 2 c ln | sin c x − cos c x | {\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|} ∫ sin c x d x cos c x + sin c x = x 2 − 1 2 c ln | sin c x + cos c x | {\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|} ∫ sin c x d x cos c x − sin c x = − x 2 − 1 2 c ln | sin c x − cos c x | {\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|} ∫ cos c x d x sin c x ( 1 + cos c x ) = − 1 4 c tg 2 c x 2 + 1 2 c ln | tg c x 2 | {\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|} ∫ cos c x d x sin c x ( 1 − cos c x ) = − 1 4 c ctg 2 c x 2 − 1 2 c ln | tg c x 2 | {\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|} ∫ sin c x d x cos c x ( 1 + sin c x ) = 1 4 c ctg 2 ( c x 2 + π 4 ) + 1 2 c ln | tg ( c x 2 + π 4 ) | {\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|} ∫ sin c x d x cos c x ( 1 − sin c x ) = 1 4 c tg 2 ( c x 2 + π 4 ) − 1 2 c ln | tg ( c x 2 + π 4 ) | {\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|} ∫ sin c x cos c x d x = 1 2 c sin 2 c x {\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}}\sin ^{2}cx} ∫ sin c 1 x cos c 2 x d x = − cos ( c 1 + c 2 ) x 2 ( c 1 + c 2 ) − cos ( c 1 − c 2 ) x 2 ( c 1 − c 2 ) ( | c 1 | ≠ | c 2 | ) {\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{)}}} ∫ sin n c x cos c x d x = 1 c ( n + 1 ) sin n + 1 c x ( n ≠ 1 ) {\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}} ∫ sin c x cos n c x d x = − 1 c ( n + 1 ) cos n + 1 c x ( n ≠ 1 ) {\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{( }}n\neq 1{\mbox{)}}} ∫ sin n c x cos m c x d x = − sin n − 1 c x cos m + 1 c x c ( n + m ) + n − 1 n + m ∫ sin n − 2 c x cos m c x d x ( m , n > 0 ) {\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{)}}} ∫ sin n c x cos m c x d x = sin n + 1 c x cos m − 1 c x c ( n + m ) + m − 1 n + m ∫ sin n c x cos m − 2 c x d x ( m , n > 0 ) {\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{)}}} ∫ d x sin c x cos c x = 1 c ln | tg c x | {\displaystyle \int {\frac {dx}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\operatorname {tg} cx\right|} ∫ d x sin c x cos n c x = 1 c ( n − 1 ) cos n − 1 c x + ∫ d x sin c x cos n − 2 c x ( n ≠ 1 ) {\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{)}}} ∫ d x sin n c x cos c x = − 1 c ( n − 1 ) sin n − 1 c x + ∫ d x sin n − 2 c x cos c x ( n ≠ 1 ) {\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{)}}} ∫ sin c x d x cos n c x = 1 c ( n − 1 ) cos n − 1 c x ( n ≠ 1 ) {\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}} ∫ sin 2 c x d x cos c x = − 1 c sin c x + 1 c ln | tg ( π 4 + c x 2 ) | {\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|} ∫ sin 2 c x d x cos n c x = sin c x c ( n − 1 ) cos n − 1 c x − 1 n − 1 ∫ d x cos n − 2 c x ( n ≠ 1 ) {\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{)}}} ∫ sin n c x d x cos c x = − sin n − 1 c x c ( n − 1 ) + ∫ sin n − 2 c x d x cos c x ( n ≠ 1 ) {\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{)}}} ∫ sin n c x d x cos m c x = sin n + 1 c x c ( m − 1 ) cos m − 1 c x − n − m + 2 m − 1 ∫ sin n c x d x cos m − 2 c x ( m ≠ 1 ) {\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{)}}} ∫ sin n c x d x cos m c x = − sin n − 1 c x c ( n − m ) cos m − 1 c x + n − 1 n − m ∫ sin n − 2 c x d x cos m c x ( m ≠ n ) {\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{)}}} ∫ sin n c x d x cos m c x = sin n − 1 c x c ( m − 1 ) cos m − 1 c x − n − 1 m − 1 ∫ sin n − 1 c x d x cos m − 2 c x ( m ≠ 1 ) {\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{)}}} ∫ cos c x d x sin n c x = − 1 c ( n − 1 ) sin n − 1 c x ( n ≠ 1 ) {\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}} ∫ cos 2 c x d x sin c x = 1 c ( cos c x + ln | tg c x 2 | ) {\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\operatorname {tg} {\frac {cx}{2}}\right|\right)} ∫ cos 2 c x d x sin n c x = − 1 n − 1 ( cos c x c sin n − 1 c x ) + ∫ d x sin n − 2 c x ) ( n ≠ 1 ) {\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{)}}} ∫ cos n c x d x sin m c x = − cos n + 1 c x c ( m − 1 ) sin m − 1 c x − n − m − 2 m − 1 ∫ c o s n c x d x sin m − 2 c x ( m ≠ 1 ) {\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{)}}} ∫ cos n c x d x sin m c x = cos n − 1 c x c ( n − m ) sin m − 1 c x + n − 1 n − m ∫ c o s n − 2 c x d x sin m c x ( m ≠ n ) {\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{)}}} ∫ cos n c x d x sin m c x = − cos n − 1 c x c ( m − 1 ) sin m − 1 c x − n − 1 m − 1 ∫ c o s n − 2 c x d x sin m − 2 c x ( m ≠ 1 ) {\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{)}}} Ĕнтĕ синус тата тангенс çеç пур интегралсем
тӳрлет
∫ sin c x tg c x d x = 1 c ( ln | sec c x + tg c x | − sin c x ) {\displaystyle \int \sin cx\operatorname {tg} cx\;dx={\frac {1}{c}}(\ln |\sec cx+\operatorname {tg} cx|-\sin cx)} ∫ tg n c x d x sin 2 c x = 1 c ( n − 1 ) tg n − 1 ( c x ) ( n ≠ 1 ) {\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{)}}} Ĕнтĕ косинус тата тангенс çеç пур интегралсем
тӳрлет
∫ tg n c x d x cos 2 c x = 1 c ( n + 1 ) tg n + 1 c x ( n ≠ − 1 ) {\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{)}}} Ĕнтĕ синус тата котангенс çеç пур интегралсем
тӳрлет
∫ ctg n c x d x sin 2 c x = 1 c ( n + 1 ) ctg n + 1 c x ( n ≠ − 1 ) {\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{)}}} Ĕнтĕ косинус тата котангенс çеç пур интегралсем
тӳрлет
∫ ctg n c x d x cos 2 c x = 1 c ( 1 − n ) tg 1 − n c x ( n ≠ 1 ) {\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{)}}} Ĕнтĕ тангенс тата котангенс çеç пур интегралсем
тӳрлет
∫ tg m ( c x ) ctg n ( c x ) d x = 1 c ( m + n − 1 ) tg m + n − 1 ( c x ) − ∫ tg m − 2 ( c x ) ctg n ( c x ) d x ( m + n ≠ 1 ) {\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{)}}}
Кĕнекесем Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — 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 Интегралсен таблицисем Интегралсене шутлани