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زمان تقریبی مطالعه: 10 دقیقه
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فهرست انتگرال‌های تابع‌های هیپربولیک

در ادامه فهرستی از انتگرال تابع‌های هیپربولیک نوشته شده‌است. برای آگاهی بیشتر صفحهٔ فهرست انتگرال‌ها را نگاه کنید.

در تمامی رابطه‌ها فرض می‌شود که a ناصفر است و C ثابت انتگرال‌گیری است.

∫ sinh ⁡ a x d x = 1 a cosh ⁡ a x + C {\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C\,}
∫ cosh ⁡ a x d x = 1 a sinh ⁡ a x + C {\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C\,}
∫ sinh 2 ⁡ a x d x = 1 4 a sinh ⁡ 2 a x − x 2 + C {\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C\,}
∫ cosh 2 ⁡ a x d x = 1 4 a sinh ⁡ 2 a x + x 2 + C {\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C\,}
∫ tanh 2 ⁡ a x d x = x − tanh ⁡ a x a + C {\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C\,}
∫ sinh n ⁡ a x d x = 1 a n sinh n − 1 ⁡ a x cosh ⁡ a x − n − 1 n ∫ sinh n − 2 ⁡ a x d x (for  n > 0 ) {\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{an}}\sinh ^{n-1}ax\cosh ax-{\frac {n-1}{n}}\int \sinh ^{n-2}ax\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,}
also: ∫ sinh n ⁡ a x d x = 1 a ( n + 1 ) sinh n + 1 ⁡ a x cosh ⁡ a x − n + 2 n + 1 ∫ sinh n + 2 ⁡ a x d x (for  n < 0 ,  n ≠ − 1 ) {\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{a(n+1)}}\sinh ^{n+1}ax\cosh ax-{\frac {n+2}{n+1}}\int \sinh ^{n+2}ax\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}
∫ cosh n ⁡ a x d x = 1 a n sinh ⁡ a x cosh n − 1 ⁡ a x + n − 1 n ∫ cosh n − 2 ⁡ a x d x (for  n > 0 ) {\displaystyle \int \cosh ^{n}ax\,dx={\frac {1}{an}}\sinh ax\cosh ^{n-1}ax+{\frac {n-1}{n}}\int \cosh ^{n-2}ax\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,}
also: ∫ cosh n ⁡ a x d x = − 1 a ( n + 1 ) sinh ⁡ a x cosh n + 1 ⁡ a x − n + 2 n + 1 ∫ cosh n + 2 ⁡ a x d x (for  n < 0 ,  n ≠ − 1 ) {\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax-{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}
∫ d x sinh ⁡ a x = 1 a ln ⁡ | tanh ⁡ a x 2 | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}
also: ∫ d x sinh ⁡ a x = 1 a ln ⁡ | cosh ⁡ a x − 1 sinh ⁡ a x | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\,}
also: ∫ d x sinh ⁡ a x = 1 a ln ⁡ | sinh ⁡ a x cosh ⁡ a x + 1 | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\,}
also: ∫ d x sinh ⁡ a x = 1 a ln ⁡ | cosh ⁡ a x − 1 cosh ⁡ a x + 1 | + C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\,}
∫ d x cosh ⁡ a x = 2 a arctan ⁡ e a x + C {\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}
also: ∫ d x cosh ⁡ a x = 1 a arctan ⁡ ( sinh ⁡ a x ) + C {\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {1}{a}}\arctan(\sinh ax)+C\,}
∫ d x sinh n ⁡ a x = − cosh ⁡ a x a ( n − 1 ) sinh n − 1 ⁡ a x − n − 2 n − 1 ∫ d x sinh n − 2 ⁡ a x (for  n ≠ 1 ) {\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ d x cosh n ⁡ a x = sinh ⁡ a x a ( n − 1 ) cosh n − 1 ⁡ a x + n − 2 n − 1 ∫ d x cosh n − 2 ⁡ a x (for  n ≠ 1 ) {\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ cosh n ⁡ a x sinh m ⁡ a x d x = cosh n − 1 ⁡ a x a ( n − m ) sinh m − 1 ⁡ a x + n − 1 n − m ∫ cosh n − 2 ⁡ a x sinh m ⁡ a x d x (for  m ≠ n ) {\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,}
also: ∫ cosh n ⁡ a x sinh m ⁡ a x d x = − cosh n + 1 ⁡ a x a ( m − 1 ) sinh m − 1 ⁡ a x + n − m + 2 m − 1 ∫ cosh n ⁡ a x sinh m − 2 ⁡ a x d x (for  m ≠ 1 ) {\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,}
also: ∫ cosh n ⁡ a x sinh m ⁡ a x d x = − cosh n − 1 ⁡ a x a ( m − 1 ) sinh m − 1 ⁡ a x + n − 1 m − 1 ∫ cosh n − 2 ⁡ a x sinh m − 2 ⁡ a x d x (for  m ≠ 1 ) {\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,}
∫ sinh m ⁡ a x cosh n ⁡ a x d x = sinh m − 1 ⁡ a x a ( m − n ) cosh n − 1 ⁡ a x + m − 1 n − m ∫ sinh m − 2 ⁡ a x cosh n ⁡ a x d x (for  m ≠ n ) {\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,}
also: ∫ sinh m ⁡ a x cosh n ⁡ a x d x = sinh m + 1 ⁡ a x a ( n − 1 ) cosh n − 1 ⁡ a x + m − n + 2 n − 1 ∫ sinh m ⁡ a x cosh n − 2 ⁡ a x d x (for  n ≠ 1 ) {\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
also: ∫ sinh m ⁡ a x cosh n ⁡ a x d x = − sinh m − 1 ⁡ a x a ( n − 1 ) cosh n − 1 ⁡ a x + m − 1 n − 1 ∫ sinh m − 2 ⁡ a x cosh n − 2 ⁡ a x d x (for  n ≠ 1 ) {\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ x sinh ⁡ a x d x = 1 a x cosh ⁡ a x − 1 a 2 sinh ⁡ a x + C {\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C\,}
∫ x cosh ⁡ a x d x = 1 a x sinh ⁡ a x − 1 a 2 cosh ⁡ a x + C {\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C\,}
∫ x 2 cosh ⁡ a x d x = − 2 x cosh ⁡ a x a 2 + ( x 2 a + 2 a 3 ) sinh ⁡ a x + C {\displaystyle \int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C\,}
∫ tanh ⁡ a x d x = 1 a ln ⁡ cosh ⁡ a x + C {\displaystyle \int \tanh ax\,dx={\frac {1}{a}}\ln \cosh ax+C\,}
∫ coth ⁡ a x d x = 1 a ln ⁡ | sinh ⁡ a x | + C {\displaystyle \int \coth ax\,dx={\frac {1}{a}}\ln |\sinh ax|+C\,}
∫ tanh n ⁡ a x d x = − 1 a ( n − 1 ) tanh n − 1 ⁡ a x + ∫ tanh n − 2 ⁡ a x d x (for  n ≠ 1 ) {\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ coth n ⁡ a x d x = − 1 a ( n − 1 ) coth n − 1 ⁡ a x + ∫ coth n − 2 ⁡ a x d x (for  n ≠ 1 ) {\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}
∫ sinh ⁡ a x sinh ⁡ b x d x = 1 a 2 − b 2 ( a sinh ⁡ b x cosh ⁡ a x − b cosh ⁡ b x sinh ⁡ a x ) + C (for  a 2 ≠ b 2 ) {\displaystyle \int \sinh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}\,}
∫ cosh ⁡ a x cosh ⁡ b x d x = 1 a 2 − b 2 ( a sinh ⁡ a x cosh ⁡ b x − b sinh ⁡ b x cosh ⁡ a x ) + C (for  a 2 ≠ b 2 ) {\displaystyle \int \cosh ax\cosh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}\,}
∫ cosh ⁡ a x sinh ⁡ b x d x = 1 a 2 − b 2 ( a sinh ⁡ a x sinh ⁡ b x − b cosh ⁡ a x cosh ⁡ b x ) + C (for  a 2 ≠ b 2 ) {\displaystyle \int \cosh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\sinh bx-b\cosh ax\cosh bx)+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}\,}
∫ sinh ⁡ ( a x + b ) sin ⁡ ( c x + d ) d x = a a 2 + c 2 cosh ⁡ ( a x + b ) sin ⁡ ( c x + d ) − c a 2 + c 2 sinh ⁡ ( a x + b ) cos ⁡ ( c x + d ) + C {\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C\,}
∫ sinh ⁡ ( a x + b ) cos ⁡ ( c x + d ) d x = a a 2 + c 2 cosh ⁡ ( a x + b ) cos ⁡ ( c x + d ) + c a 2 + c 2 sinh ⁡ ( a x + b ) sin ⁡ ( c x + d ) + C {\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C\,}
∫ cosh ⁡ ( a x + b ) sin ⁡ ( c x + d ) d x = a a 2 + c 2 sinh ⁡ ( a x + b ) sin ⁡ ( c x + d ) − c a 2 + c 2 cosh ⁡ ( a x + b ) cos ⁡ ( c x + d ) + C {\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C\,}
∫ cosh ⁡ ( a x + b ) cos ⁡ ( c x + d ) d x = a a 2 + c 2 sinh ⁡ ( a x + b ) cos ⁡ ( c x + d ) + c a 2 + c 2 cosh ⁡ ( a x + b ) sin ⁡ ( c x + d ) + C {\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C\,}

منابع

مشارکت‌کنندگان ویکی‌پدیا. «List of integrals of hyperbolic functions». در دانشنامهٔ ویکی‌پدیای انگلیسی، بازبینی‌شده در ۳۱ اوت ۲۰۱۱.

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