$\int \frac{(1 + x^{2}) d x}{(1 - x^{2}) \sqrt{1 + x^{4}}}$

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Solution

$\begin{matrix} H e r e g i v e n, I = \frac{1 - x^{2}}{(1 + x^{2}) \sqrt{1 + x^{4}}} d x I = \frac{1 + \frac{1}{x^{2}}}{(\frac{1}{x} - x) \sqrt{x^{2} + \frac{1}{x^{2}}}} d x N o w, p u t : x - \frac{1}{x} = - u \Rightarrow (1 + \frac{1}{x^{2}}) d x = d u h e n c e, i n t e g r a l i s . . . . . . . . . . . . . . . I = \frac{d u}{- u \sqrt{u^{2} + 2}} d x A g a i n, P u t u = \sqrt{2} tan (z) d u = \sqrt{2} s e c^{2} (z) d z N o w - i n t e g r a l i s : . . . . . . . . . I = \frac{\sqrt{2} s e c^{2} (z) d z}{- 2 t a n (z) \sqrt{2} s e c (z)} ∴ I = \frac{- 1}{2} \int | c s c (z) - c o t (z) | + C \end{matrix}$

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