1-decompose-inside-C-x-and-R-x-F-1-x-2-x-1-2-2-calculate-0-dx-x-2-x-1-2-

Question Number 82433 by mathmax by abdo last updated on 21/Feb/20

1) d e c o m p o s e i n s i d e C (x) a n d R (x) F = \frac{1}{{(x^{2} + x + 1)}^{2}}

2) c a l c u l a t e \int_{0}^{\infty} \frac{d x}{{(x^{2} + x + 1)}^{2}}

Commented by mathmax by abdo last updated on 24/Feb/20

x^{2} + x + 1 = 0 \to Δ = 1 - 4 = - 3 \Rightarrow z_{1} = \frac{- 1 + i \sqrt{3}}{2} = e^{i \frac{2 π}{3}}

z_{2} = e^{- \frac{i 2 π}{3}} \Rightarrow F = \frac{1}{{(x - e^{\frac{i 2 π}{3}})}^{2} {(x - e^{- \frac{i 2 π}{3}})}^{2}} = \frac{a}{x - e^{\frac{i 2 π}{3}}} + \frac{b}{{(x - e^{\frac{i 2 π}{3}})}^{2}}

+ \frac{c}{(x - e^{- i \frac{2 π}{3}})} + \frac{d}{{(x - e^{- i \frac{2 π}{3}})}^{2}}

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

d = \frac{1}{{(2 i s i n (\frac{2 π}{3}))}^{2}} = - \frac{1}{3 i} \Rightarrow F (x) = \frac{a}{(x - e^{\frac{i 2 π}{3}})} - \frac{1}{3 i {(x - e^{\frac{i 2 π}{3}})}^{2}}

+ \frac{c}{x - e^{- \frac{i 2 π}{3}}} - \frac{1}{3 i {(x - e^{- \frac{i 2 π}{3}})}^{2}}

{l i m}_{x \to + \infty} x F (x) = 0 = a + c \Rightarrow c = - a

F (0) = 1 = - {a e}^{- \frac{i 2 π}{3}} - \frac{1}{3 i} e^{- \frac{i 4 π}{3}} + {a e}^{\frac{i 2 π}{3}} - \frac{1}{3 i} e^{\frac{i 4 π}{3}}

= 2 i a s i n (\frac{2 π}{3}) - \frac{1}{3 i} (2 c o s (\frac{4 π}{3})) = 2 i a \times \frac{\sqrt{3}}{2} - \frac{2}{3 i} (- \frac{1}{2})

= a i \sqrt{3} + \frac{1}{3 i} = a i \sqrt{3} - \frac{i}{3} = 1 \Rightarrow a i \sqrt{3} = 1 + \frac{i}{3} \Rightarrow

a = \frac{1}{i \sqrt{3}} (1 + \frac{i}{3}) = \frac{1}{i \sqrt{3}} + \frac{1}{3 \sqrt{3}} = \frac{1}{3 \sqrt{3}} - \frac{i}{\sqrt{3}} a n d c = - \frac{1}{3 \sqrt{3}} + \frac{i}{\sqrt{3}} \Rightarrow

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

+ \frac{i}{3 {(x - e^{- i \frac{2 π}{3}})}^{2}}