calculate-I-0-arctan-x-x-3-dx-

Question Number 159592 by mnjuly1970 last updated on 19/Nov/21

c a l c u l a t e :

I := \int_{0}^{\infty} {(\frac{a r c t a n (x)}{x})}^{3} d x = ?

Answered by mindispower last updated on 19/Nov/21

b y a r t = \frac{3}{2} \int_{0}^{\infty} \frac{{a r c t a n}^{2} (x)}{x^{2} (1 + x^{2})} d x

= 3 \int_{0}^{\infty} \frac{a r c t a n (x)}{x (1 + x^{2})} d x - \frac{1}{2} . {(\frac{π}{2})}^{3}

= 3 \int_{0}^{\frac{π}{2}} \frac{x c o s (x)}{s i n (x)} - \frac{π^{3}}{16}

- 3 \int_{0}^{\frac{π}{2}} l n (s i n (x)) d x - \frac{π^{3}}{16}

\frac{3 π}{2} l n (2) - \frac{π^{3}}{16}

Commented by mnjuly1970 last updated on 19/Nov/21

t h a n k s a l o t s i r p o w e r

Commented by mindispower last updated on 19/Nov/21

w i t h e p l e a s u r s i r