advanced-calculus-evsluate-I-0-xsin-2x-x-2-4-dx-hint-0-cos-px-x-2-1-dx-pi-2-e-p-p-gt-0-m

Question Number 117088 by mnjuly1970 last updated on 09/Oct/20

\dots a d v a n c e d c a l c u l u s \dots

e v s l u a t e ::

I = \int_{0}^{\infty} \frac{x s i n (2 x)}{x^{2} + 4} d x = ?

h i n t :

Φ = \int_{0}^{\infty} \frac{c o s (p x)}{x^{2} + 1} d x = \frac{π}{2} e^{- p} (p > 0)

. m . n 1970

Answered by mnjuly1970 last updated on 09/Oct/20

Φ = \int_{0}^{\infty} \frac{c o s (p x)}{x^{2} + 1} \underset{r e s p e c t t o p}{\overset{(d i f f b o t h s i d e s)}{=}} \frac{π}{2} e^{- p}

= - \int_{0}^{\infty} \frac{x s i n (p x)}{x^{2} + 1} d x = - \frac{π}{2} e^{- p}

= \int_{0}^{\infty} \frac{x s i n (p x)}{x^{2} + 1} d x = \frac{π}{2} e^{p}

I \overset{x = 2 t}{=} \int_{0}^{\infty} \frac{t s i n (4 t)}{t^{2} + 1} d t \overset{Φ}{=} \frac{π}{2} e^{- 4}

. . m . n . 1970 \dots

Answered by Bird last updated on 10/Oct/20

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

= I m (\int_{- \infty}^{+ \infty} \frac{{x e}^{2 i x}}{x^{2} + 4} d x) l e t

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

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

= 2 i π \times \frac{2 i e^{- 4}}{4 i} = \frac{- 4 π e^{- 4}}{4 i} = i π e^{- 4}

\Rightarrow 2 I = π e^{- 4} \Rightarrow I = \frac{π}{2} e^{- 4}

Commented by mnjuly1970 last updated on 10/Oct/20

g r a t e f u l m r b i r d

v e r y n i c e a s a l w a y s . .