calculate-x-sinh-1-x-dx-where-sinh-1-x-is-the-inverse-hyperbolic-sine-function-

Question Number 99114 by M±th+et+s last updated on 18/Jun/20

c a l c u l a t e :

\int \sqrt{x} {s i n h}^{- 1} (x) d x

w h e r e {s i n h}^{- 1} (x) i s t h e i n v e r s e h y p e r b o l i c

s i n e f u n c t i o n

Answered by Rio Michael last updated on 19/Jun/20

By parts

let u = \sinh^{- 1} x and v^{'} = \sqrt{x}

\Rightarrow u^{'} = \frac{1}{\sqrt{1 + x^{2}}} and v = \frac{2 x^{\frac{3}{2}}}{3}

\Rightarrow \int \sqrt{x} \sinh^{- 1} x d x = [\frac{2 x^{3 / 2}}{3} \sinh^{- 1} x] - \frac{2}{3} \int \frac{x^{3 / 2}}{\sqrt{x^{2} + 1}} d x

to be continued \dots . .