calculste-I-cos-x-n-1-x-2-dx-with-from-R-and-n-integr-natural-2-find-the-vslue-of-cos-3-x-9-1-x-2-dx-

Question Number 39787 by abdo mathsup 649 cc last updated on 10/Jul/18

c a l c u l s t e I_{λ} = \int_{- \infty}^{+ \infty} \frac{c o s (λ x^{n})}{1 + x^{2}} d x w i t h

λ f r o m R a n d n i n t e g r n a t u r a l

2) f i n d t h e v s l u e o f \int_{- \infty}^{+ \infty} \frac{c o s (3 x^{9})}{1 + x^{2}} d x .

Commented by abdo mathsup 649 cc last updated on 11/Jul/18

1) w e h a v e I_{λ} = R e (\int_{- \infty}^{+ \infty} \frac{e^{i λ x^{n}}}{1 + x^{2}} d x) l e t

φ (z) = \frac{e^{i λ z^{n}}}{1 + z^{2}}

φ (z) = \frac{e^{i λ z^{n}}}{(z - i) (z + i)} \Rightarrow

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

R e s (φ, i) = {l i m}_{z \to i} (z - i) φ (z)

= {l i m}_{z \to i} \frac{e^{i λ z^{n}}}{z + i} = \frac{e^{λ i^{n + 1}}}{2 i} = \frac{e^{λ {c o s (\frac{(n + 1) π}{2} + i s i n (\frac{(n + 1) π}{2})}}}{2 i}

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π e^{λ c o s (\frac{(n + 1) π}{2})} \frac{c o s) (λ s i n (\frac{(n + 1) π}{2})) + i s i n (λ s i n (\frac{(n + 1) π}{2}))}{2 i}

\Rightarrow I_{λ} = π e^{λ c o s (\frac{(n + 1) π}{2})} c o s {λ s i n (\frac{(n + 1) π}{2})} .

Commented by math khazana by abdo last updated on 12/Jul/18

2) l e t t a k e λ = 3 a n d n = 9 w e g e t

\int_{- \infty}^{+ \infty} \frac{c o s (3 x^{9})}{1 + x^{2}} d x = π e^{3 c o s (5 π)} c o s {3 s i n (5 π)}

= π e^{- 3} = \frac{π}{e^{3}} .