INTEGRALS CONTAINING Cosh(ax)

anshulpatel

friends i hav come across many JEE papers in which these types of integration sums r not been able to b solved by the students ................ plzz refer the following formulae.............it will surely b useful.........do rate me if u like it.........!!!!!!!!

plzz do rate only those wch u think was useful.............!!!!!!!!!!

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.: $displaystyleintdisplaystyle rac{xdx}{sinh^n ax}=displaystyle rac{-xco... ...-2}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{xdx}{sinh^{n-2}ax}$

INTEGRALS CONTAINING Cosh(ax)

1.
2.: $displaystyleint xcosh axdx=displaystyle rac{xsinh ax}{a}-displaystyle rac{cosh ax}{a^2}$
3.: $displaystyleint x^2cosh axdx=-displaystyle rac{2xcosh ax}{a^2}+left( displaystyle rac{x^2}{a}+displaystyle rac{2}{a^3} ight) sinh ax$
4.: $displaystyleintdisplaystyle rac{cosh ax}{x}dx=ln x+displaystyle rac{(... ... rac{(ax)^4}{4cdot 4!}+displaystyle rac{(ax)^6}{6cdot 6!}+cdotcdotcdot$
5.: $displaystyleintdisplaystyle rac{cosh ax}{x^2}dx=-displaystyle rac{cosh ax}{x}+aintdisplaystyle rac{sinh ax}{x}dx$
6.: $displaystyleintdisplaystyle rac{dx}{cosh ax}=displaystyle rac{2}{a} an^{-1e^{ax}}$
7.: $displaystyleintdisplaystyle rac{x dx}{cosh ax}=displaystyle rac{1}{a^2... ...playstyle rac{(-1)^n E_{n}(ax)^{2n+2}}{(2n+2)(2n)!}+ cdotcdotcdot ight}$
where the constants E_n are the EULERS NOS..........!!!!!!
8.: $displaystyleintcosh^2 ax dx=displaystyle rac{x}{2}+displaystyle rac{sinh ax cosh ax}{2}$
9.: $displaystyleint xcosh^2 ax dx=displaystyle rac{x^2}{4}+displaystyle rac{xsinh 2ax}{4a}-displaystyle rac{cosh 2ax}{8a^2}$
10.: $displaystyleintdisplaystyle rac{dx}{cosh^2 ax}=displaystyle rac{ anh ax}{a}$
11.
12.: $displaystyleintcosh axsin px dx=displaystyle rac{asinh axsin px -pcosh axcos px}{a^2+p^2}$
13.: $displaystyleintcosh ax cos pxdx=displaystyle rac{asinh axcos px+pcosh axsin px}{a^2+p^2}$
14.
15.
16.: $displaystyleintdisplaystyle rac{x dx}{cosh ax+1}=displaystyle rac{x}{a... ...tyle rac{ax}{2}-displaystyle rac{2}{a^2}lncoshdisplaystyle rac{ax}{2}$
17.: $displaystyleintdisplaystyle rac{x dx}{cosh ax-1}=-displaystyle rac{x}{... ...tyle rac{ax}{2}+displaystyle rac{2}{a^2}lnsinhdisplaystyle rac{ax}{2}$
TAKE A BREAK.......now continue................!!!!!!!!!

18.: $displaystyleintdisplaystyle rac{dx}{(cosh ax+1)^2}=displaystyle rac{1}... ...ystyle rac{ax}{2}-displaystyle rac{1}{6a} anh^3displaystyle rac{ax}{2}$
19.: $displaystyleintdisplaystyle rac{dx}{(cosh ax-1)^2}=displaystyle rac{1}... ...ystyle rac{ax}{2}-displaystyle rac{1}{6a}coth^3displaystyle rac{ax}{2}$
20.: $displaystyleintdisplaystyle rac{dx}{p+qcosh ax}=left{ egin{array}{ll}... ...rt{p^2-q^2}}{qe^{ax}+p+displaystyle sqrt{p^2-q^2}} ight) end{array} ight.$
21.: $displaystyleintdisplaystyle rac{dx}{(p+qcosh ax)^2}=displaystyle rac{q... ...sh ax)}-displaystyle rac{p}{q^2-p^2}intdisplaystyle rac{dx}{p+qcosh ax}$
22.: $displaystyleintdisplaystyle rac{dx}{p^2-q^2cosh^2 ax}=left{ egin{arra... ...isplaystyle rac{p anh ax}{displaystyle sqrt{q^2-p^2}} end{array} ight.$
23.: $displaystyleintdisplaystyle rac{dx}{p^2+q^2cosh^2 ax}=left{ egin{arra... ...displaystyle rac{p anh ax}{displaystyle sqrt{p^2+q^2}} end{array} ight.$
24.: $displaystyleint x^m cosh ax dx=displaystyle rac{x^m sinh ax}{a}-displaystyle rac{m}{a}int x^{m-1}sinh ax dx$
25.: $displaystyleintcosh^n ax dx=displaystyle rac{cosh^{n-1}axsinh ax}{an}+displaystyle rac{n-1}{n}intcosh^{n-2} ax dx$
26.: $displaystyleintdisplaystyle rac{cosh ax}{x^n}dx=displaystyle rac{-cos... ...^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{sinh ax}{x^{n-1}}dx$
27.: $displaystyleintdisplaystyle rac{dx}{cosh^n ax}=displaystyle rac{sinh ... ...n-1}ax}+displaystyle rac{n-2}{n-1}intdisplaystyle rac{dx}{cosh^{n-2}ax}$
28.: $displaystyleintdisplaystyle rac{x dx}{cosh^n ax}=displaystyle rac{xsi... ...2}ax}+displaystyle rac{n-2}{n-1}intdisplaystyle rac{x dx}{cosh^{n-2}ax}$

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INTEGRALS CONTAINING Cosh(ax)