1-find-x-arctan-x-dx-2-find-the-value-of-0-1-x-arctan-x-dx-

Question Number 47852 by maxmathsup by imad last updated on 15/Nov/18

1) f i n d \int x a r c t a n (x) d x

2) f i n d t h e v a l u e o f \int_{0}^{1} x a r c t a n (x) d x

Commented by maxmathsup by imad last updated on 16/Nov/18

1) l e t A = \int x a r c t a n (x) d x b y p a r t s

A = \frac{x^{2}}{2} a r c t a n (x) - \int \frac{x^{2}}{2} \frac{d x}{1 + x^{2}} = \frac{x^{2}}{2} a r c t a n (x) - \frac{1}{2} \int \frac{x^{2} + 1 - 1}{x^{2} + 1} d x

= \frac{x^{2}}{2} a r c t a n (x) - \frac{x}{2} + \frac{1}{2} a r c t a n (x) + c .

2) \int_{0}^{1} x a r c t a n (x) d x = {[\frac{x^{2}}{2} a r c t a n (x) - \frac{x}{2} + \frac{a r c t a n (x)}{2}]}_{0}^{1}

= \frac{π}{8} - \frac{1}{2} + \frac{π}{8} = \frac{π}{4} - \frac{1}{2} .