حساب کاربری
​
تغیر مسیر یافته از - فهرست انتگرال‌های توابع هایپربولیک
زمان تقریبی مطالعه: 10 دقیقه
لینک کوتاه

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.

آخرین نظرات
کلیه حقوق این تارنما متعلق به فرا دانشنامه ویکی بین است.