0-sinx-2-1-x-4-dx-0-4009-prove-that-

Question Number 85999 by M±th+et£s last updated on 26/Mar/20

\int_{0}^{\infty} \frac{{s i n x}^{2}}{1 + x^{4}} d x = 0.4009

p r o v e t h a t

Commented by MJS last updated on 26/Mar/20

\sin (x^{2}) or {(\sin x)}^{2} ?

Commented by john santu last updated on 26/Mar/20

m a y b e \sin (x^{2})

Commented by M±th+et£s last updated on 26/Mar/20

s i n (x^{2})

Commented by mathmax by abdo last updated on 26/Mar/20

I = \int_{0}^{\infty} \frac{s i n (x^{2})}{x^{4} + 1} d x \Rightarrow 2 I = \int_{- \infty}^{+ \infty} \frac{s i n (x^{2})}{x^{4} + 1} d x = I m (\int_{- \infty}^{+ \infty} \frac{e^{{i x}^{2}}}{x^{4} + 1} d x)

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

= \frac{e^{{i z}^{2}}}{(z - \sqrt{i}) (z + \sqrt{i}) (z - \sqrt{- i}) (z + \sqrt{- i})} = \frac{e^{{i z}^{2}}}{(z - e^{\frac{i π}{4}}) (z + e^{\frac{i π}{4}}) (z - e^{- \frac{i π}{4}}) (z + e^{- \frac{i π}{4}})}

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, e^{\frac{i π}{4}}) + R e s (φ, - e^{- \frac{i π}{4}})}

R e s (φ, e^{\frac{i π}{4}}) = \frac{e^{{i e}^{\frac{i π}{2}}}}{2 e^{\frac{i π}{4}} (2 i)} = \frac{e^{- 1} e^{- \frac{i π}{4}}}{4 i}

R e s (φ, - e^{- \frac{i π}{4}}) = \frac{e^{i (- i)}}{- 2 e^{- \frac{i π}{4}} (- 2 i)} = \frac{e e^{\frac{i π}{4}}}{4 i} \Rightarrow

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π {\frac{e^{- 1}}{4 i} e^{- \frac{i π}{4}} + \frac{e}{4 i} e^{\frac{i π}{4}}}

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

= \frac{π}{2} {\frac{e + e^{- 1}}{\sqrt{2}} + i (\frac{e - e^{- 1}}{\sqrt{2}})} \Rightarrow 2 I = \frac{π}{2 \sqrt{2}} (e - e^{- 1}) \Rightarrow

I = \frac{π}{4 \sqrt{2}} (e - e^{- 1})

Commented by mind is power last updated on 27/Mar/20

S i r f o r θ \in [\frac{π}{2}, π] C = {R e}^{i θ}

\frac{e^{{i z}^{2}}}{1 + z^{4}} = \frac{e^{{i R}^{2} (c o s (2 θ) + i s i n (2 θ))}}{1 + R^{4} e^{4 i θ}} = \frac{e^{- R^{2} s i n (2 θ)}}{1 + R^{4} e^{4 i θ}}, s i n (2 θ) < 0

You can't use 'macro parameter character #' in math mode