let-F-x-x-x-2-arctan-xt-x-2-t-2-dt-calculate-F-x-

Question Number 67974 by mathmax by abdo last updated on 02/Sep/19

l e t F (x) = \int_{x}^{x^{2}} \frac{a r c t a n (x t)}{x^{2} + t^{2}} d t c a l c u l a t e F^{^{'}} (x) .

Commented by mathmax by abdo last updated on 03/Sep/19

w e h a v e g (x, t) = \frac{a r c t a n (x t)}{x^{2} + t^{2}}, u (x) = x a n d v (x) = x^{2}

w e u s e t h e f o r m u l a e F^{^{'}} (x) = \int_{u (x)}^{v (x)} \frac{\partial g}{\partial x} (x, t) d t + v^{^{'}} g (x, v)

- u^{^{'}} g (x, u) = \int_{x}^{x^{2}} \frac{\partial g}{\partial x} (x, t) d t + 2 x \times \frac{a r c t a n (x^{3})}{x^{2} + x^{4}}

- \frac{a r c t a n (x^{2})}{2 x^{2}}

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

Commented by mathmax by abdo last updated on 03/Sep/19

\Rightarrow F^{^{'}} (x) = \int_{x}^{x^{2}} (\frac{t}{(1 + x^{2} t^{2}) (x^{2} + t^{2})} - \frac{2 x a r c t a n (x t)}{{(x^{2} + t^{2})}^{2}}) d t

+ \frac{2 x a r c t a n (x^{3})}{x^{2} + x^{4}} - \frac{a r c t a n (x^{2})}{2 x^{2}} \dots . b e c o n t i n u e d \dots