0-1-cot-1-1-x-x-2-dx-

Question Number 85280 by Jakir Sarif Mondal last updated on 20/Mar/20

\underset{0}{\int^{1}} \cot^{- 1} (1 - x + x^{2}) d x =

Answered by mind is power last updated on 20/Mar/20

= \int_{0}^{1} {t a n}^{-} (\frac{1}{1 - x (1 - x)}) d x

= \int_{0}^{1} {t a n}^{-} (\frac{1 - x + x}{1 - x (1 - x)}) d x = \int_{0}^{1} ({t a n}^{-} (1 - x) + {t a n}^{-} (x)) d x

= 2 \int_{0}^{1} {t a n}^{-} (x) d x

= 2 {[{x t a n}^{-} (x) - l n (1 + x^{2})]}_{0}^{1_{}}

= \frac{π}{2} - l n (2)