✨Danh sách tích phân với hàm hypebolic

Dưới đây là danh sách tích phân với hàm hypebolic.

: $\backslash int\backslash sinh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\backslash cosh\; cx$

: $\backslash int\backslash cosh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\backslash sinh\; cx$

: $\backslash int\backslash sinh^2\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{4c\}\backslash sinh\; 2cx\; -\; \backslash frac\{x\}\{2\}$

: $\backslash int\backslash cosh^2\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{4c\}\backslash sinh\; 2cx\; +\; \backslash frac\{x\}\{2\}$

: $\backslash int\backslash sinh^n\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{cn\}\backslash sinh^\{n-1\}\; cx\backslash cosh\; cx\; -\; \backslash frac\{n-1\}\{n\}\backslash int\backslash sinh^\{n-2\}\; cx\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n>0\backslash mbox\{)\}$

: hay:

: $\backslash int\backslash cosh^n\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{cn\}\backslash sinh\; cx\backslash cosh^\{n-1\}\; cx\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash cosh^\{n-2\}\; cx\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n>0\backslash mbox\{)\}$

: hay:

: $\backslash int\backslash frac\{dx\}\{\backslash sinh\; cx\}\; =\; \backslash frac\{1\}\{c\}\; \backslash ln\backslash left|\backslash tanh\backslash frac\{cx\}\{2\}\backslash right|$

: hay: $\backslash int\backslash frac\{dx\}\{\backslash sinh\; cx\}\; =\; \backslash frac\{1\}\{c\}\; \backslash ln\backslash left|\backslash frac\{\backslash cosh\; cx\; -\; 1\}\{\backslash sinh\; cx\}\backslash right|$

: hay: $\backslash int\backslash frac\{dx\}\{\backslash sinh\; cx\}\; =\; \backslash frac\{1\}\{c\}\; \backslash ln\backslash left|\backslash frac\{\backslash sinh\; cx\}\{\backslash cosh\; cx\; +\; 1\}\backslash right|$

: hay: $\backslash int\backslash frac\{dx\}\{\backslash sinh\; cx\}\; =\; \backslash frac\{1\}\{c\}\; \backslash ln\backslash left|\backslash frac\{\backslash cosh\; cx\; -\; 1\}\{\backslash cosh\; cx\; +\; 1\}\backslash right|$

: $\backslash int\backslash frac\{dx\}\{\backslash cosh\; cx\}\; =\; \backslash frac\{2\}\{c\}\; \backslash arctan\; e^\{cx\}$

: $\backslash int\backslash frac\{dx\}\{\backslash sinh^n\; cx\}\; =\; \backslash frac\{\backslash cosh\; cx\}\{c(n-1)\backslash sinh^\{n-1\}\; cx\}-\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash sinh^\{n-2\}\; cx\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{\backslash cosh^n\; cx\}\; =\; \backslash frac\{\backslash sinh\; cx\}\{c(n-1)\backslash cosh^\{n-1\}\; cx\}+\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cosh^\{n-2\}\; cx\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{\backslash cosh^n\; cx\}\{\backslash sinh^m\; cx\}\; dx\; =\; \backslash frac\{\backslash cosh^\{n-1\}\; cx\}\{c(n-m)\backslash sinh^\{m-1\}\; cx\}\; +\; \backslash frac\{n-1\}\{n-m\}\backslash int\backslash frac\{\backslash cosh^\{n-2\}\; cx\}\{\backslash sinh^m\; cx\}\; dx\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; n\backslash mbox\{)\}$

: hay: $\backslash int\backslash frac\{\backslash cosh^n\; cx\}\{\backslash sinh^m\; cx\}\; dx\; =\; -\backslash frac\{\backslash cosh^\{n+1\}\; cx\}\{c(m-1)\backslash sinh^\{m-1\}\; cx\}\; +\; \backslash frac\{n-m+2\}\{m-1\}\backslash int\backslash frac\{\backslash cosh^n\; cx\}\{\backslash sinh^\{m-2\}\; cx\}\; dx\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; 1\backslash mbox\{)\}$

: hay: $\backslash int\backslash frac\{\backslash cosh^n\; cx\}\{\backslash sinh^m\; cx\}\; dx\; =\; -\backslash frac\{\backslash cosh^\{n-1\}\; cx\}\{c(m-1)\backslash sinh^\{m-1\}\; cx\}\; +\; \backslash frac\{n-1\}\{m-1\}\backslash int\backslash frac\{\backslash cosh^\{n-2\}\; cx\}\{\backslash sinh^\{m-2\}\; cx\}\; dx\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{\backslash sinh^m\; cx\}\{\backslash cosh^n\; cx\}\; dx\; =\; \backslash frac\{\backslash sinh^\{m-1\}\; cx\}\{c(m-n)\backslash cosh^\{n-1\}\; cx\}\; +\; \backslash frac\{m-1\}\{m-n\}\backslash int\backslash frac\{\backslash sinh^\{m-2\}\; cx\}\{\backslash cosh^n\; cx\}\; dx\; \backslash qquad\backslash mbox\{(\}m\backslash neq\; n\backslash mbox\{)\}$

: hay: $\backslash int\backslash frac\{\backslash sinh^m\; cx\}\{\backslash cosh^n\; cx\}\; dx\; =\; \backslash frac\{\backslash sinh^\{m+1\}\; cx\}\{c(n-1)\backslash cosh^\{n-1\}\; cx\}\; +\; \backslash frac\{m-n+2\}\{n-1\}\backslash int\backslash frac\{\backslash sinh^m\; cx\}\{\backslash cosh^\{n-2\}\; cx\}\; dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: hay: $\backslash int\backslash frac\{\backslash sinh^m\; cx\}\{\backslash cosh^n\; cx\}\; dx\; =\; -\backslash frac\{\backslash sinh^\{m-1\}\; cx\}\{c(n-1)\backslash cosh^\{n-1\}\; cx\}\; +\; \backslash frac\{m-1\}\{n-1\}\backslash int\backslash frac\{\backslash sinh^\{m\; -2\}\; cx\}\{\backslash cosh^\{n-2\}\; cx\}\; dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\; x\backslash sinh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\; x\backslash cosh\; cx\; -\; \backslash frac\{1\}\{c^2\}\backslash sinh\; cx$

: $\backslash int\; x\backslash cosh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\; x\backslash sinh\; cx\; -\; \backslash frac\{1\}\{c^2\}\backslash cosh\; cx$

: $\backslash int\; \backslash tanh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\backslash ln|\backslash cosh\; cx|$

: $\backslash int\; \backslash coth\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\backslash ln|\backslash sinh\; cx|$

: $\backslash int\; \backslash tanh^n\; cx\backslash ,dx\; =\; -\backslash frac\{1\}\{c(n-1)\}\backslash tanh^\{n-1\}\; cx+\backslash int\backslash tanh^\{n-2\}\; cx\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\; \backslash coth^n\; cx\backslash ,dx\; =\; -\backslash frac\{1\}\{c(n-1)\}\backslash coth^\{n-1\}\; cx+\backslash int\backslash coth^\{n-2\}\; cx\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\; \backslash sinh\; bx\; \backslash sinh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{b^2-c^2\}\; (b\backslash sinh\; cx\; \backslash cosh\; bx\; -\; c\backslash cosh\; cx\; \backslash sinh\; bx)\; \backslash qquad\backslash mbox\{(\}b^2\backslash neq\; c^2\backslash mbox\{)\}$

: $\backslash int\; \backslash cosh\; bx\; \backslash cosh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{b^2-c^2\}\; (b\backslash sinh\; bx\; \backslash cosh\; cx\; -\; c\backslash sinh\; cx\; \backslash cosh\; bx)\; \backslash qquad\backslash mbox\{(\}b^2\backslash neq\; c^2\backslash mbox\{)\}$

: $\backslash int\; \backslash cosh\; bx\; \backslash sinh\; cx\backslash ,dx\; =\; \backslash frac\{1\}\{b^2-c^2\}\; (b\backslash sinh\; bx\; \backslash sinh\; cx\; -\; c\backslash cosh\; bx\; \backslash cosh\; cx)\; \backslash qquad\backslash mbox\{(\}b^2\backslash neq\; c^2\backslash mbox\{)\}$

: $\backslash int\; \backslash sinh\; (ax+b)\backslash sin\; (cx+d)\backslash ,dx\; =\; \backslash frac\{a\}\{a^2+c^2\}\backslash cosh(ax+b)\backslash sin(cx+d)-\backslash frac\{c\}\{a^2+c^2\}\backslash sinh(ax+b)\backslash cos(cx+d)$

: $\backslash int\; \backslash sinh\; (ax+b)\backslash cos\; (cx+d)\backslash ,dx\; =\; \backslash frac\{a\}\{a^2+c^2\}\backslash cosh(ax+b)\backslash cos(cx+d)+\backslash frac\{c\}\{a^2+c^2\}\backslash sinh(ax+b)\backslash sin(cx+d)$

: $\backslash int\; \backslash cosh\; (ax+b)\backslash sin\; (cx+d)\backslash ,dx\; =\; \backslash frac\{a\}\{a^2+c^2\}\backslash sinh(ax+b)\backslash sin(cx+d)-\backslash frac\{c\}\{a^2+c^2\}\backslash cosh(ax+b)\backslash cos(cx+d)$

: $\backslash int\; \backslash cosh\; (ax+b)\backslash cos\; (cx+d)\backslash ,dx\; =\; \backslash frac\{a\}\{a^2+c^2\}\backslash sinh(ax+b)\backslash cos(cx+d)+\backslash frac\{c\}\{a^2+c^2\}\backslash cosh(ax+b)\backslash sin(cx+d)$