Indefinite Integrals with $\frac{1}{x}$

1/x
$\int\arg ch\dfrac{a}{x}dx=\begin{cases} \dfrac{x\arg ch\dfrac{a}{x}+\arcsin\dfrac{x}{a}}{x\arg ch\dfrac{a}{x}-\arcsin\dfrac{x}{a}} & .\end{cases} +C$
$\int x\arg ch\dfrac{a}{x} dx=\begin{cases} \dfrac{\frac{1}{2}x^{2}\arg ch\dfrac{a}{x}-\dfrac{1}{2}a\sqrt{a^{2}-x^{2}}}{\frac{1}{2}x^{2}\arg ch\dfrac{a}{x}+\dfrac{1}{2}a\sqrt{a^{2}-x^{2}}} & .\end{cases} +C$
$\int\dfrac{\arg ch\dfrac{a}{x}}{x}dx=\begin{cases} \dfrac{\dfrac{-\frac{1}{2}\ln(\dfrac{a}{x})\ln(\dfrac{4a}{x})}{2}-\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}-\cdots}{\dfrac{\frac{1}{2}\ln(\dfrac{a}{x})\ln(\dfrac{4a}{x})}{2}+\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots} & .\end{cases} +C$
$\int\arg sh\dfrac{a}{x}dx=x\arg sh\dfrac{a}{x}\pm\arg sh\dfrac{x}{a} +C$
$\int x\arg sh\dfrac{a}{x} dx=\dfrac{x^{2}}{2}\arg sh\dfrac{a}{x}\pm\dfrac{1}{2}a\sqrt{a^{2}+x^{2}} +C$
$\int\dfrac{\arg sh\dfrac{a}{x}}{x}dx=\Biggl\{\begin{array}{c} \dfrac{\frac{1}{2}\ln(\dfrac{x}{a})\ln(\dfrac{4a}{x})}{2}+\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots\\ \dfrac{\frac{1}{2}\ln(\dfrac{x}{a})\ln(\dfrac{4a}{x})}{2}-\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots\\ -\frac{a}{x}+\dfrac{(\dfrac{a}{x})^{3}}{2\cdot3\cdot3}-\dfrac{1\cdot3(\dfrac{a}{x})^{5}}{2\cdot4\cdot5\cdot5}+\cdots\end{array}$
$\int x^{m}\arg sh\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg sh\dfrac{x}{a}-\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}+a^{2}}}dx$
$\int x^{m}\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{x}{a}-\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}-a^{2}}}dx}{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{x}{a}+\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}-a^{2}}}dx} & .\end{cases}$
$\int x^{m}\arg th\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg th\dfrac{x}{a}-\dfrac{a}{m+1} \int\dfrac{x^{m+1}}{a^{2}-x^{2}}dx$
$\int x^{m}\arg coth\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg coth\dfrac{x}{a}-\dfrac{a}{m+1} \int\dfrac{x^{m+1}}{a^{2}-x^{2}}dx$
$\int x^{m}\arg ch\dfrac{a}{x} dx=\begin{cases} \dfrac{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{a}{x}+\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{a^{2}-x^{2}}}dx}{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{a}{x}-\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{a^{2}-x^{2}}}dx} & .\end{cases}$
$\int x^{m}\arg sh\dfrac{a}{x} dx=\dfrac{x^{m+1}}{m+1}\arg sh\dfrac{x}{a}\pm\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{x^{2}+a^{2}}}dx$

## Alumni Liaison

Recent Math PhD now doing a post-doctorate at UC Riverside.

Kuei-Nuan Lin