ខាងក្រោមនេះជាតារាងអាំងតេក្រាលនៃអនុគមន៍អ៊ីពែបូលីក៖ គ្រប់រូបមន្ត a {\displaystyle a\,} ជាចំនួនថេរ និង c {\displaystyle c\,} ជាចំនួនថេរអាំងតេក្រាល។ ∫ 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 {\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\,} ចំពោះ n > 0 {\displaystyle n>0\,} ∫ 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 {\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\,} ចំពោះ n < 0 , n ≠ − 1 {\displaystyle n<0\,,\,n\neq -1\,} ∫ 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 {\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\,} ចំពោះ n > 0 {\displaystyle n>0\,} ∫ 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 {\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\,} ចំពោះ n < 0 , n ≠ − 1 {\displaystyle n<0\,,\,n\neq -1\,} ∫ 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\,} ∫ 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\,} ∫ 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\,} ∫ 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\,} ∫ 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 {\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}}\,} ចំពោះ n ≠ 1 {\displaystyle n\neq 1\,} ∫ 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 {\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}}\,} ចំពោះ n ≠ 1 {\displaystyle n\neq 1\,} ∫ 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 {\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\,} ចំពោះ m ≠ n {\displaystyle m\neq n\,} ∫ 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 {\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\,} ចំពោះ m ≠ 1 {\displaystyle m\neq 1\,} ∫ 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 {\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\,} ចំពោះ m ≠ 1 {\displaystyle m\neq 1\,} ∫ 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 m − n ∫ sinh m − 2 a x cosh n a x d x {\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}{m-n}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\,} ចំពោះ m ≠ n {\displaystyle m\neq n\,} ∫ 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 {\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\,} ចំពោះ n ≠ 1 {\displaystyle n\neq 1\,} ∫ 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 {\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\,} ចំពោះ n ≠ 1 {\displaystyle n\neq 1\,} ∫ 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 {\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\,} ចំពោះ n ≠ 1 {\displaystyle n\neq 1\,} ∫ coth n a x d x = − 1 a ( n − 1 ) coth n − 1 a x + ∫ coth n − 2 a x d x {\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\,} ចំពោះ n ≠ 1 {\displaystyle n\neq 1\,} ∫ 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 {\displaystyle \int \sinh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\,} ចំពោះ a 2 ≠ b 2 {\displaystyle a^{2}\neq b^{2}\,} ∫ 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 {\displaystyle \int \cosh ax\cosh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\,} ចំពោះ a 2 ≠ b 2 {\displaystyle a^{2}\neq b^{2}\,} ∫ 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 {\displaystyle \int \cosh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\sinh bx-b\cosh ax\cosh bx)+C\,} ចំពោះ a 2 ≠ b 2 {\displaystyle a^{2}\neq b^{2}\,} ∫ 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\,} Remove adsមើលផងដែរ អាំងតេក្រាលកំនត់ អាំងតេក្រាលឌុប អាំងតេក្រាលត្រីគុណ អាំងតេក្រាលខ្សែកោង អាំងតេក្រាលផ្ទៃ Loading related searches...Wikiwand - on Seamless Wikipedia browsing. 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