INTEGRALS CONTAINING Sinh(ax)

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INTEGRALS CONTAINING Sinh(ax)


1.
2.: $displaystyleintsinh axdx=displaystyle rac{acosh ax}{a}-displaystyle rac{sinh ax}{a^2}$
3.: $displaystyleint x^2sinh axdx=left(displaystyle rac{x^2}{a}+displaystyle rac{2}{a^3} ight)cosh ax-displaystyle rac{2x}{a^2}sinh ax$
4.: $displaystyleintdisplaystyle rac{sinh ax}{x}dx=ax+displaystyle rac{(ax)^3}{3cdot 3!}+displaystyle rac{(ax)^5}{5cdot 5!}+cdotcdotcdot$
5.: $displaystyleintdisplaystyle rac{sinh ax}{x^2}dx=-displaystyle rac{sinh ax}{x}+aintdisplaystyle rac{cosh ax}{x}dx$
6.
7.: $displaystyleintdisplaystyle rac{xdx}{sinh ax}=displaystyle rac{1}{a^2}... ...tyle rac{2(-1)^n(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)!}+cdotcdotcdot ight}$
8.: $displaystyleintsinh^2 axdx=displaystyle rac{sinh axcosh ax}{2a}-displaystyle rac{x}{2}$
9.: $displaystyleint xsinh^2 axdx=displaystyle rac{xsinh 2ax}{4a}-displaystyle rac{cosh 2ax}{8a^2}-displaystyle rac{x^2}{4}$
10.: $displaystyleintdisplaystyle rac{dx}{sinh^2 ax}=-displaystyle rac{coth ax}{a}$
11.
12.: $displaystyleintsinh axsinh pxdx=displaystyle rac{acosh axsin px-psinh axcos px}{a^2 + p^2}$
13.: $displaystyleintsinh ax cos pxdx=displaystyle rac{acosh ax cos px +psinh axsin px}{a^2 + p^2}$

14.: $displaystyleintdisplaystyle rac{dx}{p+qsinh ax}=displaystyle rac{1}{a... ...+p-displaystyle sqrt{p^2+q^2}}{qe^{ax}+p+displaystyle sqrt{p^2+q^2}} ight)$
15.: $displaystyleintdisplaystyle rac{dx}{(p+qsinh ax)^2}=displaystyle rac{-... ...nh ax)}+displaystyle rac{p}{p^2+q^2}intdisplaystyle rac{dx}{p+qsinh ax}$
16.: $displaystyleintdisplaystyle rac{dx}{p^2+q^2sinh^2 ax}=left{ egin{arra... ...2} anh ax}{p-displaystyle sqrt{p^2-q^2} anh ax} ight) end{array} ight.$
17.: $displaystyleintdisplaystyle rac{dx}{p^2-q^2sinh^2 ax}=displaystyle rac... ...laystyle sqrt{p^2+q^2} anh ax}{p-displaystyle sqrt{p^2+q^2} anh ax} ight)$
18.: $displaystyleint x^msinh ax dx=displaystyle rac{x^mcosh ax}{a}-displaystyle rac{m}{a}int x^{m-1}cosh axdx$
19.: $displaystyleintsinh^n axdx=displaystyle rac{sinh^{n-1}axcosh ax}{an}-displaystyle rac{n-1}{n}intsinh^{n-2}axdx$
20.: $displaystyleintdisplaystyle rac{sinh ax}{x^n}dx=displaystyle rac{-sin... ...^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{cosh ax}{x^{n-1}}dx$
21.: $displaystyleintdisplaystyle rac{dx}{sinh^n ax}=displaystyle rac{-cosh... ...n-1}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{dx}{sinh^{n-2}ax}$
22.: $displaystyleintdisplaystyle rac{xdx}{sinh^n ax}=displaystyle rac{-xco... ...-2}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{xdx}{sinh^{n-2}ax}$

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INTEGRALS CONTAINING Sinh(ax)