Evaluate-0-dx-x-4-2x-2-cos-1-0-lt-lt-pi-

Question Number 2210 by Yozzi last updated on 08/Nov/15

E v a l u a t e

\int_{0}^{\infty} \frac{d x}{x^{4} + 2 x^{2} c o s α + 1} (0 < α < π) .

Commented by 123456 last updated on 09/Nov/15

- \frac{1}{2} ı cosec α (\frac{\tan^{- 1} \frac{x}{\sqrt{e^{- ı α}}}}{\sqrt{e^{- ı α}}} - \frac{\tan^{- 1} \frac{x}{\sqrt{e^{ı α}}}}{\sqrt{e^{ı α}}})

c . m

Commented by prakash jain last updated on 09/Nov/15

\frac{1}{x^{4} + 2 x^{2} c o s α + \cos^{2} α + 1 - \cos^{2} α}

= \frac{1}{{(x^{2} + \cos α)}^{2} + \sin^{2} α}

= \frac{1}{(x^{2} + \cos α - i \sin α) (x^{2} + \cos α + i \sin α)}

= \frac{1}{(x^{2} + e^{- i α}) (x^{2} + e^{i α})}

Answered by prakash jain last updated on 09/Nov/15

\frac{1}{(x^{2} + e^{i α}) (x^{2} + e^{- i α})}

= \frac{1}{(e^{i α} - e^{- i α})} (\frac{1}{x^{2} + e^{- i α}} - \frac{1}{x^{2} + e^{i α}})

Integrating \int \frac{1}{x^{2} + a^{2}} = \frac{1}{a} \tan^{- 1} \frac{x}{a}

= \frac{1}{2 \sin i α} [\frac{1}{\sqrt{e^{- i α}}} \tan^{- 1} \frac{x}{\sqrt{e^{- i α}}} - \frac{1}{\sqrt{e^{i α}}} \tan^{- 1} \frac{x}{\sqrt{e^{i α}}}]