Question

INTEGRATE..............

Answer 1

$I= _ { - ∞}∫^{ + ∞} \frac{x^{2} }{ {x}^{4} + 1 } dx$

$I= \frac{1}{2} \: _{ -∞ }∫^{ + ∞} \frac{1 - \frac{1}{x} }{(x + \frac{1}{x} ) ^{2} - 2} dx + \frac{1}{2} _{ - ∞} ∫ ^{ +∞} \frac{1 + \frac{1}{x} }{(x - \frac{1}{x} ) ^{2} + 2} dx$

$I= \frac{ 1}{2 \sqrt{2} } \tan^{ - 1} ( \frac{x - \frac{1}{x} }{ \sqrt{2} } ) | _{ - ∞} ^{ +∞ } - \frac{1}{2 \sqrt{2} } \tanh ^{ - 1} ( \frac{x + \frac{1}{ x } }{\sqrt{2} } ) | _{ - ∞} ^{ + ∞}$

$I= \frac{\pi}{ \sqrt{2} }$

Answer 2

Answer:

your answer is given in the attachment