Analytical Integration of Arccosec x

We can integrate  
\[cosec^{-1} x\]
  by parts by writing  
\[cosec^{-1}x=1 \times cosec^{-1}x\]
.
Let  
\[u=cosec^{-1}x \rightarrow cosecu=x \rightarrow -cosecucotu \frac{du}{dx}=- 1\]
  then
\[\frac{du}{dx}= - \frac{1}{cosecucotu} =-\frac{1}{x \sqrt{cosec^2 u-1}}=-\frac{1}{x \sqrt{x^2-1}}\]
.
\[\frac{dv}{dx}=1 \rightarrow v=x\]

\[\begin{equation} \begin{aligned} \int 1 \times cosec^{-1}xdx &= x cosec^{-1}x - \int -x \times \frac{1}{x \sqrt{x^2-1}}dx \\ &= x cosec^{-1}x+ \int \frac{1}{\sqrt{x^2-1}}dx \\ &= x cosec^{-1}x +ln(x+\sqrt{x^2-1})+c \end{aligned} \end{equation}\]

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