0-JS-dx-0-pi-2-sin-x-4-sin-2-x-4-sin-2-x-2-dx-

Question Number 111082 by john santu last updated on 02/Sep/20

[\int_{0}^{\infty} J S d x]

\underset{0}{\int^{\frac{π}{2}}} \frac{\sin (x) (4 + \sin^{2} (x))}{{(4 - \sin^{2} (x))}^{2}} d x ?

Commented by mathdave last updated on 02/Sep/20

u s i n g {k i n g}^{'} s p r o p e r t y i t w i l l k i l l i t

Answered by Sarah85 last updated on 02/Sep/20

t = \cos x \Rightarrow d x = - \frac{d t}{\sin x} leads to

\int \frac{t^{2} - 5}{{(t^{2} + 3)}^{2}} d t = \int \frac{d t}{t^{2} + 3} - 8 \int \frac{d t}{{(t^{2} + 3)}^{2}} =

= - \frac{\arctan \frac{t}{\sqrt{3}}}{3 \sqrt{3}} - \frac{4 t}{3 (t^{2} + 3)}

\dots

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