find-the-value-of-dx-x-4-x-2-3-

Question Number 56827 by maxmathsup by imad last updated on 24/Mar/19

f i n d t h e v a l u e o f \int_{- \infty}^{+ \infty} \frac{d x}{x^{4} - x^{2} + 3}

Commented by maxmathsup by imad last updated on 25/Mar/19

r e s i d u s m e t h o d l e t φ (z) = \frac{1}{z^{4} - z^{2} + 3} p o l e s o f φ

z^{4} - z^{2} + 3 = 0 \Rightarrow t^{2} - t + 3 = 0 w i t h t = z^{2}

Δ = 1 - 12 = - 11 \Rightarrow t_{1} = \frac{1 + i \sqrt{11}}{2} a n d t_{2} = \frac{1 - i \sqrt{11}}{2} \Rightarrow φ (z) = \frac{1}{(z^{2} - t_{1}) (z^{2} - t_{2})}

= \frac{1}{(z - \sqrt{t_{1}}) (z + \sqrt{t_{1}}) (z - \sqrt{t_{2}}) (z + \sqrt{t_{2}})} w e h a v e

∣ t_{1} ∣ = \sqrt{\frac{1}{4} + \frac{11}{4}} = \sqrt{3} \Rightarrow t_{1} = \sqrt{3} (\frac{1}{2 \sqrt{3}} + \frac{i \sqrt{11}}{2 \sqrt{3}}) = r e^{i θ} \Rightarrow c o s θ = \frac{1}{2 \sqrt{3}} a n d s i n θ = \frac{\sqrt{11}}{2 \sqrt{3}}

\Rightarrow t a n θ = \sqrt{11} \Rightarrow θ = a r c t a n (\sqrt{11}) \Rightarrow t_{1} = \sqrt{3} e^{i a r c t a n (\sqrt{11})}

\Rightarrow φ (z) = \frac{1}{(z - \sqrt{\sqrt{3}} e^{\frac{i}{2} a r c t a n (\sqrt{11})}) (z + \sqrt{\sqrt{3}} e^{\frac{i}{2} a r c t a n (\sqrt{11})}) (z - \sqrt{\sqrt{3}} e^{- \frac{i}{2} a r c t a n (\sqrt{11})}) (z + \sqrt{\sqrt{3}} e^{- \frac{i}{2} a r c t a n (\sqrt{11})})}

r e s i d u s t h e o r e m g i v e \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π {R e s (φ, \sqrt{\sqrt{3}} e^{\frac{i}{2} a r c t a n (\sqrt{11})} + R e s (φ, - \sqrt{\sqrt{3}} e^{- \frac{i}{2} a r c t a n (\sqrt{11})})}

R e s (φ, \sqrt{\sqrt{3}} e^{\frac{i}{2} a r c t a n (\sqrt{11})}) = \frac{1}{2 \sqrt{\sqrt{3}} e^{\frac{i}{2} a r c t a n (\sqrt{11})} 2 i \sqrt{3} s i n (\frac{a r c t a n (\sqrt{11})}{2}) 2 c o s (\frac{a r c t a n (\sqrt{11})}{2})}

= \frac{e^{- \frac{i}{2} a r c t a n (\sqrt{11})}}{4 i \sqrt{3} \sqrt{\sqrt{3}} s i n (a r c t a n (\sqrt{11}))}

R e s (φ, - \sqrt{\sqrt{3}} e^{- \frac{i}{2} a r c t a n (\sqrt{11})}) = \frac{1}{- 2 \sqrt{\sqrt{3}} e^{- \frac{i}{2} a r c t a n (\sqrt{11})} (- \sqrt{\sqrt{3}}) 2 c o s (\frac{a r c t a n (\sqrt{11})}{2}) 2 i \sqrt{\sqrt{3}} s i n (\frac{a r c t a n (\sqrt{11})}{2})}

= \frac{e^{\frac{i}{2} a r c t a n (\sqrt{11})}}{4 i \sqrt{3} \sqrt{\sqrt{3}} s i n (a r c t a n (\sqrt{11}))} \Rightarrow

\int_{- \infty}^{+ \infty} φ (z) d z = \frac{2 i π}{4 i \sqrt{3} \sqrt{\sqrt{3}} s i n (a r c t a n (\sqrt{11}))} {e^{\frac{i}{2} a r c t a n (\sqrt{11})} + e^{- \frac{i}{2} a r c t a n (\sqrt{11)}}}

= \frac{π}{2 \sqrt{3} \sqrt{\sqrt{3}} s i n (a r c t a n (\sqrt{11})} . 2 c o s (\frac{a r c t a n (\sqrt{11})}{2}) \Rightarrow

I = \frac{π}{\sqrt{3} \sqrt{\sqrt{3}}} \frac{c o s (\frac{a r c t a n (\sqrt{11})}{2})}{2 s i n (\frac{a r c t a n (\sqrt{11})}{2}) c o s (\frac{a r c t a n (\sqrt{11})}{2})}

I = \frac{π}{2 \sqrt{3} \sqrt{\sqrt{3}} s i n (\frac{a r c t a n (\sqrt{11})}{2})} .

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Mar/19

\frac{1}{x^{4} - x^{2} + 3}

= \frac{\frac{1}{x^{2}}}{x^{2} - 1 + \frac{3}{x^{2}}}

= \frac{1}{2 \sqrt{3}} (\frac{\frac{2 \sqrt{3}}{x^{2}}}{x^{2} - 1 + \frac{3}{x^{2}}})

= \frac{1}{2 \sqrt{3}} (\frac{1 + \frac{\sqrt{3}}{x^{2}} - (1 - \frac{\sqrt{3}}{x^{2}})}{x^{2} + \frac{3}{x^{2}} - 1})

= \frac{1}{2 \sqrt{3}} (\frac{d (x - \frac{\sqrt{3}}{x})}{{(x - \frac{\sqrt{3}}{x})}^{2} + 2 \sqrt{3} - 1}) - \frac{1}{2 \sqrt{3}} (\frac{d (x + \frac{\sqrt{3}}{x})}{{(x + \frac{\sqrt{3}}{x})}^{2} - 2 \sqrt{3} - 1})

\frac{1}{2 \sqrt{3}} \int_{- \infty}^{+ \infty} \frac{d (x - \frac{\sqrt{3}}{x})}{{(x - \frac{\sqrt{3}}{x})}^{2} + {(\sqrt{2 \sqrt{3} - 1})}^{2}} \leftarrow I_{1}

\frac{1}{2 \sqrt{3}} \times \frac{1}{\sqrt{2 \sqrt{3} - 1}} \times ∣ {t a n}^{- 1} (\frac{x - \frac{\sqrt{3}}{x}}{\sqrt{2 \sqrt{3} - 1}}) ∣_{- \infty}^{+ \infty}

= \frac{1}{2 \sqrt{3}} \times \frac{1}{\sqrt{2 \sqrt{3} - 1}} \times {(\frac{π}{2}) - (\frac{- π}{2})}

= \frac{π}{2 \sqrt{3}} \times \frac{1}{\sqrt{2 \sqrt{3} - 1}} \leftarrow v a l u e o f I_{1}

\frac{1}{2 \sqrt{3}} \int_{- \infty}^{+ \infty} \frac{d (x + \frac{\sqrt{3}}{x})}{{(x + \frac{\sqrt{3}}{x})}^{2} - (2 \sqrt{3} + 1)} \leftarrow I_{2}

\frac{1}{2 \sqrt{3}} \times \frac{1}{2 (\sqrt{2 \sqrt{3} + 1})} ∣ l n (\frac{x + \frac{\sqrt{3}}{x} - \sqrt{2 \sqrt{3} + 1}}{x + \frac{\sqrt{3}}{x} + \sqrt{2 \sqrt{3} + 1}}) ∣_{- \infty}^{+ \infty}

n o w ∣ l n (\frac{x + \frac{a}{x} - b}{x + \frac{a}{x} + b}) ∣_{- \infty}^{+ \infty}

=∣ l n (\frac{x^{2} + a - b x}{x^{2} + a + b x}) ∣_{- \infty}^{+ \infty}

=∣ l n (\frac{1 + \frac{a}{x^{2}} - \frac{b}{x}}{1 + \frac{a}{x^{2}} + \frac{b}{x}}) ∣_{- \infty}^{+ \infty} = 0

s o I_{2} = 0

h e n c e a n s w e r i s

\frac{π}{2 \sqrt{3}} \times \frac{1}{\sqrt{2 \sqrt{3} - 1}} t h i s i s t h e a n s w e r