Таблиця інтегралів гіперболічних функцій

Це список інтегралів (первісних функцій) гіперболічних функцій. Для повнішого списку інтегралів дивись Таблиця інтегралів.

У всіх цих формурах під a мається на увазі ненульова константа, C означає константу інтегрування.

${\displaystyle \int {\text{sh}}\;ax\;dx={\frac {{\text{ch}}\;ax}{a}}+C\,}$

${\displaystyle \int {\text{ch}}\;ax\;dx={\frac {{\text{sh}}\;ax}{a}}+C\,}$

${\displaystyle \int {\text{sh}}^{2}ax\,dx={\frac {{\text{sh}}\;2ax}{4a}}-{\frac {x}{2}}+C\,}$

${\displaystyle \int {\text{ch}}^{2}ax\,dx={\frac {{\text{sh}}\;2ax}{4a}}+{\frac {x}{2}}+C\,}$

${\displaystyle \int {\text{th}}^{2}ax\,dx=x-{\frac {{\text{th}}\;ax}{a}}+C\,}$

${\displaystyle \int {\text{sh}}^{n}ax\,dx={\frac {{\text{sh}}^{n-1}ax\;{\text{ch}}\;ax}{an}}-{\frac {n-1}{n}}\int {\text{sh}}^{n-2}ax\,dx\qquad (}$ для ${\displaystyle n>0)\,}$

також: ${\displaystyle \int {\text{sh}}^{n}ax\,dx={\frac {{\text{sh}}^{n+1}ax\;{\text{ch}}\;ax}{a(n+1)}}-{\frac {n+2}{n+1}}\int {\text{sh}}^{n+2}ax\,dx\qquad (}$ для ${\displaystyle n<0{\mbox{, }}n\neq -1)\,}$

${\displaystyle \int {\text{ch}}^{n}ax\,dx={\frac {{\text{sh}}\;ax\;{\text{ch}}^{n-1}ax}{an}}+{\frac {n-1}{n}}\int {\text{ch}}^{n-2}ax\,dx\qquad (}$ для ${\displaystyle n>0)\,}$

також: ${\displaystyle \int {\text{ch}}^{n}ax\,dx=-{\frac {{\text{sh}}\;ax\;{\text{ch}}^{n+1}ax}{a(n+1)}}-{\frac {n+2}{n+1}}\int {\text{ch}}^{n+2}ax\,dx\qquad (}$ для ${\displaystyle n<0{\mbox{, }}n\neq -1)\,}$

${\displaystyle \int {\frac {dx}{{\text{sh}}\;ax}}={\frac {1}{a}}\ln \left|{\text{th}}{\frac {ax}{2}}\right|+C\,}$

також: ${\displaystyle \int {\frac {dx}{{\text{sh}}\;ax}}={\frac {1}{a}}\ln \left|{\frac {{\text{ch}}\;ax-1}{{\text{sh}}\;ax}}\right|+C\,}$

також: ${\displaystyle \int {\frac {dx}{{\text{sh}}\;ax}}={\frac {1}{a}}\ln \left|{\frac {{\text{sh}}\;ax}{{\text{ch}}\;ax+1}}\right|+C\,}$

також: ${\displaystyle \int {\frac {dx}{{\text{sh}}\;ax}}={\frac {1}{a}}\ln \left|{\frac {{\text{ch}}\;ax-1}{{\text{ch}}\;ax+1}}\right|+C\,}$

${\displaystyle \int {\frac {dx}{{\text{ch}}\;ax}}={\frac {2}{a}}{\text{arctg}}\;e^{ax}+C\,}$

${\displaystyle \int {\frac {dx}{{\text{sh}}^{n}ax}}=-{\frac {{\text{ch}}\;ax}{a(n-1){\text{sh}}^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{{\text{sh}}^{n-2}ax}}\qquad (}$ для ${\displaystyle n\neq 1)\,}$

${\displaystyle \int {\frac {dx}{{\text{ch}}^{n}ax}}={\frac {{\text{sh}}\;ax}{a(n-1){\text{ch}}^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{{\text{ch}}^{n-2}ax}}\qquad (}$ для ${\displaystyle n\neq 1)\,}$

${\displaystyle \int {\frac {{\text{ch}}^{n}ax}{{\text{sh}}^{m}ax}}dx={\frac {{\text{ch}}^{n-1}ax}{a(n-m){\text{sh}}^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {{\text{ch}}^{n-2}ax}{{\text{sh}}^{m}ax}}dx\qquad (}$ для ${\displaystyle m\neq n)\,}$

також: ${\displaystyle \int {\frac {{\text{ch}}^{n}ax}{{\text{sh}}^{m}ax}}dx=-{\frac {{\text{ch}}^{n+1}ax}{a(m-1){\text{sh}}^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {{\text{ch}}^{n}ax}{{\text{sh}}^{m-2}ax}}dx\qquad (}$ для ${\displaystyle m\neq 1)\,}$

також: ${\displaystyle \int {\frac {{\text{ch}}^{n}ax}{{\text{sh}}^{m}ax}}dx=-{\frac {{\text{ch}}^{n-1}ax}{a(m-1){\text{sh}}^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {{\text{ch}}^{n-2}ax}{{\text{sh}}^{m-2}ax}}dx\qquad (}$ для ${\displaystyle m\neq 1)\,}$

${\displaystyle \int {\frac {{\text{sh}}^{m}ax}{{\text{ch}}^{n}ax}}dx={\frac {{\text{sh}}^{m-1}ax}{a(m-n){\text{ch}}^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {{\text{sh}}^{m-2}ax}{{\text{ch}}^{n}ax}}dx\qquad (}$ для ${\displaystyle m\neq n)\,}$

також: ${\displaystyle \int {\frac {{\text{sh}}^{m}ax}{{\text{ch}}^{n}ax}}dx={\frac {{\text{sh}}^{m+1}ax}{a(n-1){\text{ch}}^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {{\text{sh}}^{m}ax}{{\text{ch}}^{n-2}ax}}dx\qquad (}$ для ${\displaystyle n\neq 1)\,}$

також: ${\displaystyle \int {\frac {{\text{sh}}^{m}ax}{{\text{ch}}^{n}ax}}dx=-{\frac {{\text{sh}}^{m-1}ax}{a(n-1){\text{ch}}^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {{\text{sh}}^{m-2}ax}{{\text{ch}}^{n-2}ax}}dx\qquad (}$ для ${\displaystyle n\neq 1)\,}$

${\displaystyle \int x\;{\text{sh}}\;ax\,dx={\frac {x\;{\text{ch}}\;ax}{a}}-{\frac {{\text{sh}}\;ax}{a^{2}}}+C\,}$

${\displaystyle \int x\;{\text{ch}}\;ax\,dx={\frac {x\;{\text{sh}}\;ax}{a}}-{\frac {{\text{ch}}\;ax}{a^{2}}}+C\,}$

${\displaystyle \int x^{2}{\text{ch}}\;ax\,dx=-{\frac {2x\;{\text{ch}}\;ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right){\text{sh}}\;ax+C\,}$

${\displaystyle \int {\text{th}}\;ax\,dx={\frac {\ln |{\text{ch}}\;ax|}{a}}+C\,}$

${\displaystyle \int {\text{cth}}\;ax\,dx={\frac {\ln |{\text{sh}}\;ax|}{a}}+C\,}$

${\displaystyle \int {\text{th}}^{n}ax\,dx=-{\frac {{\text{th}}^{n-1}ax}{a(n-1)}}+\int {\text{th}}^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}$

${\displaystyle \int {\text{cth}}^{n}ax\,dx=-{\frac {{\text{cth}}^{n-1}ax}{a(n-1)}}+\int {\text{cth}}^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}$

${\displaystyle \int {\text{sh}}\;ax\;{\text{sh}}\;bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\;{\text{sh}}\;bx\;{\text{ch}}\;ax-b\;{\text{ch}}\;bx\;{\text{sh}}\;ax)+C\qquad (}$ для ${\displaystyle a^{2}\neq b^{2})\,}$

${\displaystyle \int {\text{ch}}\;ax\;{\text{ch}}\;bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\;{\text{sh}}\;ax\;{\text{ch}}\;bx-b\;{\text{sh}}\;bx\;{\text{ch}}\;ax)+C\qquad (}$ для ${\displaystyle a^{2}\neq b^{2})\,}$

${\displaystyle \int {\text{ch}}\;ax\;{\text{sh}}\;bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\;{\text{sh}}\;ax\;{\text{sh}}\;bx-b\;{\text{ch}}\;ax\;{\text{ch}}\;bx)+C\qquad (}$ для ${\displaystyle a^{2}\neq b^{2})\,}$

${\displaystyle \int {\text{sh}}(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}{\text{ch}}(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}{\text{sh}}(ax+b)\cos(cx+d)+C\,}$

${\displaystyle \int {\text{sh}}(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}{\text{ch}}(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}{\text{sh}}(ax+b)\sin(cx+d)+C\,}$

${\displaystyle \int {\text{ch}}(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}{\text{sh}}(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}{\text{ch}}(ax+b)\cos(cx+d)+C\,}$

${\displaystyle \int {\text{ch}}(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}{\text{sh}}(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}{\text{ch}}(ax+b)\sin(cx+d)+C\,}$

Джерела

• Гиперболичесике функции — интегралы // Таблицы интегралов и другие математические формулы / пер. с англ. Н. В. Леви ; под ред. . — М. : Наука, 1978. — С. 134-140. (рос.)