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Question Number 48720 by Abdo msup. last updated on 27/Nov/18

c a l c u l a t e \int_{0}^{\infty} \frac{x^{2} - 2 c o s x + 1}{x^{4} + x^{2} + 1} d x

Commented by Abdo msup. last updated on 28/Nov/18

l e t I = \int_{0}^{\infty} \frac{x^{2} - 2 c o s x + 1}{x^{4} + x^{2} + 1} d x \Rightarrow

2 I = \int_{- \infty}^{+ \infty} \frac{x^{2} + 1}{x^{4} + x^{2} + 1} d x - \int_{- \infty}^{+ \infty} \frac{2 c o s x}{x^{4} + x^{2} + 1} d x = H - K

l e t f i n d H l e t φ (z) = \frac{z^{2} + 1}{z^{4} + z^{2} + 1} p o l e s o f φ ?

z^{4} + z^{2} + 1 = 0 \Rightarrow t^{2} + t + 1 = 0 (t = z^{2})

Δ = - 3 = {(i \sqrt{3})}^{2} \Rightarrow z_{1} = \frac{- 1 + i \sqrt{3}}{2} = e^{\frac{i 2 π}{3}} a n d z_{2} = e^{- \frac{i 2 π}{3}}

z^{2} = e^{\frac{i 2 π}{3}} \Rightarrow z = \bar{+} e^{\frac{i π}{3}}

z^{2} = e^{- \frac{i 2 π}{3}} \Rightarrow z = \bar{+} e^{- \frac{i π}{3}} \Rightarrow

φ (z) = \frac{z^{2} + 1}{(z - e^{\frac{i π}{3}}) (z + e^{\frac{i π}{3}}) (z - e^{- \frac{i π}{3}}) (z + e^{- \frac{i π}{3}})}

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π {R e s (φ, e^{\frac{i π}{3}}) + R e s (φ, - e^{- \frac{i π}{3}})

R e s (φ, e^{\frac{i π}{3}}) = \frac{e^{2 \frac{i π}{3}} + 1}{2 e^{\frac{i π}{3}} (2 i s i n (\frac{π}{3}) (2 c o s (\frac{π}{3}))}

= \frac{e^{\frac{i π}{3}} + e^{- \frac{i π}{3}}}{8 i s i n (\frac{π}{3}) c o s (\frac{π}{3})} = \frac{2 c o s (\frac{π}{3})}{8 i s i n (\frac{π}{3}) c o s (\frac{π}{3})} = \frac{1}{4 i \frac{\sqrt{3}}{2}} = \frac{1}{2 i \sqrt{3}}

R e s (φ, - e^{- \frac{i π}{3}}) = \frac{e^{- \frac{2 i π}{3}} + 1}{- 2 c o s (\frac{π}{3}) (2 i s i n (\frac{π}{3})) (- 2 e^{- \frac{i π}{3}})}

= \frac{e^{\frac{i π}{3}} + e^{\frac{- i π}{3}}}{8 i c o s (\frac{π}{3}) s i n (\frac{π}{3})} = \frac{2 c o s (\frac{π}{3})}{8 i c o s (\frac{π}{3}) s i n (\frac{π}{3})}

= \frac{1}{4 i s i n (\frac{π}{3})} = \frac{1}{4 i \frac{\sqrt{3}}{2}} = \frac{1}{2 i \sqrt{3}} \Rightarrow

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π {\frac{1}{2 i \sqrt{3}} + \frac{1}{2 i \sqrt{3}}} = \frac{2 π}{\sqrt{3}} = H

Commented by Abdo msup. last updated on 28/Nov/18

l e t f i n d K

K = \int_{- \infty}^{+ \infty} \frac{2 c o s x}{x^{4} + x^{2} + 1} d x = R e (\int_{- \infty}^{+ \infty} \frac{2 e^{i x}}{x^{4} + x^{2} + 1} d x)

l e t W (z) = \frac{2 e^{i z}}{z^{4} + z^{2} + 1} \Rightarrow

W (z) = \frac{2 e^{i z}}{z^{4} + z^{2} + 1} \Rightarrow W (z) = \frac{2 e^{i z}}{(z - e^{\frac{i π}{3}}) (z + e^{\frac{i π}{3}}) (z - e^{- \frac{i π}{3}}) (z + e^{- \frac{i π}{3}})}

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π {R e s (φ, e^{\frac{i π}{3}}) + R e s (φ, - e^{- \frac{i π}{3}})}

R e s (φ, e^{\frac{i π}{3}}) = \frac{2 e^{i (c o s \frac{π}{3} + i s i n (\frac{π}{3}))}}{2 e^{\frac{i π}{3}} (2 i s i n (\frac{π}{3})) 2 c o s (\frac{π}{3})}

= \frac{e^{i (\frac{1}{2} + i \frac{\sqrt{3}}{2})}}{4 i s i n (\frac{π}{3}) c o s (\frac{π}{3})} e^{- \frac{i π}{3}} = \frac{e^{- \frac{\sqrt{3}}{2}} e^{i (\frac{1}{2} - \frac{π}{3})}}{4 i \frac{\sqrt{3}}{2} . \frac{1}{2}}

= \frac{e^{- \frac{\sqrt{3}}{2}}}{i \sqrt{3}} e^{i (\frac{1}{2} - \frac{π}{3})}

R e s (φ, - e^{- \frac{i π}{3}}) = \frac{2 e^{i (c o s \frac{π}{3} - i s i n (\frac{π}{3}))}}{- 2 c o s (\frac{π}{3}) (2 i s i n (\frac{π}{3}) (- 2 e^{\frac{- i π}{3}})}

= \frac{e^{i (\frac{1}{2} - i \frac{\sqrt{3}}{2})}}{4 i c o s (\frac{π}{3}) s i n (\frac{π}{3})} e^{\frac{i π}{3}} = \frac{e^{\frac{\sqrt{3}}{2}} e^{i (\frac{1}{2} + \frac{π}{3})}}{4 i \frac{1}{2} \frac{\sqrt{3}}{2}} = \frac{e^{\frac{\sqrt{3}}{2}} e^{i (\frac{1}{2} + \frac{π}{3})}}{i \sqrt{3}} \Rightarrow

\int_{- \infty}^{+ \infty} W (z) d z = 2 i π \frac{1}{i \sqrt{3}} {e^{- \frac{\sqrt{3}}{2}} e^{i (\frac{1}{2} - \frac{π}{3})} + e^{\frac{\sqrt{3}}{2}} e^{i (\frac{1}{2} + \frac{π}{3} {}}

= \frac{2 π}{\sqrt{3}} {e^{- \frac{\sqrt{3}}{2}} {c o s (\frac{1}{2} - \frac{π}{3}) + i s i n (\frac{1}{2} - \frac{π}{3}) +

e^{\frac{\sqrt{3}}{2}} {c o s (\frac{1}{2} + \frac{π}{3}) + i s i n (\frac{1}{2} + \frac{π}{3})}}

K = R e (\int_{- \infty}^{+ \infty} W (z) d z)

= \frac{2 π}{\sqrt{3}} {e^{- \frac{\sqrt{3}}{2}} c o s (\frac{1}{2} - \frac{π}{3}) + e^{\frac{\sqrt{3}}{2}} c o s (\frac{1}{2} + \frac{π}{3})}

s o t h e v a l u e o f I i s d e t e r m i n e d .