calculate-1-2-1-x-arctan-1-x-2-dx-

Question Number 51990 by maxmathsup by imad last updated on 01/Jan/19

c a l c u l a t e \int_{\frac{1}{2}}^{1} x a r c t a n (\sqrt{1 - x^{2}}) d x

Answered by peter frank last updated on 01/Jan/19

b y p a r t

v = \frac{x^{2}}{2}

u = \tan^{- 1} (\sqrt{1 - x^{2}})

\frac{d u}{d x} = - \frac{2 x}{2 - x^{2}}

\frac{x^{2}}{2} \tan^{- 1} (\sqrt{1 - x^{2}}) + \int \frac{x^{3}}{2 - x^{2}} d x

\frac{x^{2}}{2} \tan^{- 1} (\sqrt{1 - x^{2}}) + \int \frac{2 x - x^{3} + 2 x}{2 - x^{2}}

\frac{x^{2}}{2} \tan^{- 1} (\sqrt{1 - x^{2}}) + l n (2 - x^{2}) + C

\dots .