Interals Involving sinh, cosh, sech

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Interals Involving cosh ax

$\int_{}^{}coshaxdx=\frac{sinhax}{a}$

$\int_{}^{}x.coshaxdx=\frac{x.sinhax}{a}-\frac{coshax}{a^2}$

$\int_{}^{}x^2.coshaxdx=\frac{-2x.coshax}{a^2}+\left(\frac{x^2}{a}+\frac{2}{a^3}\right)sinhax$

$\int_{}^{}\frac{coshaxdx}{x}=lnx+\frac{(ax)^2}{2*2!}+\frac{(ax)^4}{4*4!}+...$

$\int_{}^{}\frac{coshaxdx}{x^2}=\frac{-coshax}{x}-a.\int_{}^{}\frac{sinhaxdx}{x}$

$\int_{}^{}\frac{dx}{coshax}=\frac{2}{a}ln(tan^{-1}(e^{ax}))$

$\int_{}^{}cosh^2axdx=\frac{x}{2}+\frac{sinh2ax}{2a}$

$\int_{}^{}x.cosh^2axdx=\frac{x^2}{4}+\frac{x.sinh2ax}{4a}-\frac{cosh2ax}{8a^2}$

$\int_{}^{}\frac{dx}{cosh^2ax}=\frac{tanhax}{a}$

Integrals Involving sinh ax and cosh ax

$\int_{}^{}sinhax.coshaxdx=\frac{sinh^2ax}{2a}$

$\int_{}^{}sinh^2ax.cosh^2axdx=\frac{sinh4ax}{32a}-\frac{x}{8}$

$\int_{}^{}\frac{dx}{sinhax.coshax}=\frac{ln(tanhax)}{a}$

$\int_{}^{}\frac{dx}{sinh^2ax.coshax}=\frac{-1}{a}tan^{-1}sinhax-\frac{cschax}{a}$

$\int_{}^{}\frac{dx}{sinhax.cosh^2ax}=\frac{sechax}{a}+\frac{1}{a}ln(tanh\frac{ax}{2})$

$\int_{}^{}\frac{dx}{sinh^2ax.cosh^2ax}=\frac{-2coth(2ax)}{a}$

$\int_{}^{}\frac{sinh^2axdx}{coshax}=\frac{sinhax}{a}-\frac{1}{a}.ln(tan^{-1}sinhax)$

$\int_{}^{}\frac{cosh^2axdx}{sinhax}=\frac{coshax}{a}+\frac{1}{a}.ln(tanh\frac{ax}{2})$

Integrals Involving tanh ax

$\int_{}^{}tanhaxdx=\frac{1}{a}.ln(coshax)$

$\int_{}^{}tanh^2axdx=x-\frac{tanhax}{a}$

$\int_{}^{}tanh^3axdx=\frac{1}{a}ln(coshax)-\frac{tanh^2ax}{2a}$

$\int_{}^{}tanh^nax.sech^2axdx=\frac{tanh^{n+1}ax}{(n+1)a}$

$\int_{}^{}\frac{sech^2axdx}{tanhax}=\frac{1}{a}.ln(tanhax)$

$\int_{}^{}\frac{dx}{tanhax}=\frac{1}{a}ln(sinhax)$

$\int_{}^{}x.tanh^2axdx=\frac{x^2}{2}-\frac{x.tanhax}{a}+\frac{1}{a^2}ln(coshax)$

Integrals Involving coth ax

$\int_{}^{}cothaxdx=\frac{1}{a}ln(sinhax)$

$\int_{}^{}coth^2axdx=x-\frac{cothax}{a}$

$\int_{}^{}coth^3axdx=\frac{1}{a}ln(sinhax)-\frac{coth^2ax}{2a}$

$\int_{}^{}coth^nax.csch^2axdx=\frac{-coth^{n+1}ax}{(n+1)a}$

$\int_{}^{}\frac{csch^2axdx}{cothax}=\frac{-1}{a}ln(cothax)$

$\int_{}^{}\frac{dx}{cothax}=\frac{-1}{a}ln(coshax)$

$\int_{}^{}xcoth^2ax=\frac{x^2}{2}-\frac{x.cothax}{a}+\frac{1}{a}ln(sinhax)$

Integrals Involving sech ax

$\int_{}^{}sechaxdx=\frac{2}{a}tan^{-1}(e^{ax})$

$\int_{}^{}sech^2axdx=\frac{tanhax}{a}$

$\int_{}^{}sech^3axdx=\frac{sechax.tanhax}{2a}+\frac{1}{2a}.tan^{-1}sinhax$

$\int_{}^{}sech^nax.tanhaxdx=\frac{-sech^nax}{na}$

$\int_{}^{}\frac{dx}{sechax}=\frac{sinhax}{a}$

$\int_{}^{}x.sech^2axdx=\frac{x}{a}tanhax-\frac{1}{a^2}ln(coshax)$

Integrals Involving csch ax

$\int_{}^{}cschaxdx=\frac{-cschax.cothax}{2a}-\frac{1}{2a}ln(tan\frac{ax}{2})$

$\int_{}^{}csch^2axdx=\frac{-cothax}{a}$

$\int_{}^{}csch^3axdx=\frac{-cschax.cothax}{2a}-\frac{1}{2a}.ln(tanh\frac{ax}{2})$

$\int_{}^{}csch^nax.cothaxdx=\frac{-csch^nax}{na}$

$\int_{}^{}\frac{dx}{cschax}=\frac{coshax}{a}$

$\int_{}^{}x.csch^2ax=\frac{-x}{a}cothax+\frac{1}{a^2}ln(sinhax)$

Integrals Involving Inverse Hyperbolic Functions

$\int_{}^{}sinh^{-1}\frac{x}{a}dx=xsinh^{-1}\frac{x}{a}-\sqrt{x^2+a^2}$

$\int_{}^{}x.sinh^{-1}\frac{x}{a}dx=(\frac{x^2}{2}+\frac{a^2}{4})sinh^{-1}\frac{x}{a}-\frac{x\sqrt{x^2+a^2}}{4}$

$\int_{}^{}x^2.sinh^{-1}\frac{x}{a}dx=\frac{x^3}{3}sinh^{-1}\frac{x}{a}+\frac{(2a^2-x^2)\sqrt{x^2+a^2}}{9}$

$\int_{}^{}tanh^{-1}\frac{x}{a}dx=x.tanh^{-1}\frac{x}{a}+\frac{a}{2}ln(a^2-x^2)$

$\int_{}^{}x.tanh^{-1}\frac{x}{a}dx=\frac{ax}{2}+\frac{1}{2}(x^2-a^2)tanh^{-1}\frac{x}{a}$

$\int_{}^{}x^2.tanh^{-1}\frac{x}{a}dx=\frac{ax^2}{6}+\frac{x^3}{3}tanh^{-1}\frac{x}{a}+\frac{a^3}{6}ln(a^2-x^2)$

$\int_{}^{}\frac{tanh^{-1}\frac{x}{a}}{x}dx=\frac{x}{a}+\frac{\left(\frac{x}{a}\right)^3}{3^2}+\frac{\left(\frac{x}{a}\right)^5}{5^2}+\frac{\left(\frac{x}{a}\right)^7}{7^2}+...$

$\int_{}^{}\frac{tanh^{-1}\frac{x}{a}}{x^2}dx=\frac{-1}{x}tanh^{-1}\frac{x}{a}+\frac{1}{2a}ln\frac{x^2}{x^2-a^2}$

$\int_{}^{}coth^{-1}\frac{x}{a}dx=x.coth^{-1}\frac{x}{a}+\frac{a}{2}ln(x^2-a^2)$

$\int_{}^{}x.coth^{-1}\frac{x}{a}dx=\frac{ax}{2}+\frac{1}{2}(x^2-a^2)coth^{-1}\frac{x}{a}$

$\int_{}^{}x^2.coth^{-1}\frac{x}{a}dx=\frac{x^3}{3}coth^{-1}\frac{x}{a}+\frac{ax^2}{6}+\frac{a^3}{6}ln(x^2-a^2)$

$\int_{}^{}\frac{coth^{-1}\frac{x}{a}}{x}dx=\frac{-a}{x}+\frac{(\frac{a}{x})^3}{3^2}+\frac{(\frac{a}{x})^5}{5^2}+...$

$\int_{}^{}\frac{coth^{-1}\frac{x}{a}}{x^2}dx=\frac{-1}{x}coth^{-1}\frac{x}{a}+\frac{1}{2a}ln\frac{x^2}{x^2-a^2}$

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