let-f-a-0-cos-x-2-sin-x-2-x-2-a-2-2-dx-with-a-gt-0-1-calculate-f-a-2-find-the-values-of-0-cos-x-2-sin-x-2-x-2-1-2-dx-and-0-cos-x-2-sin-x-2-

Question Number 64970 by mathmax by abdo last updated on 23/Jul/19

l e t f (a) = \int_{0}^{\infty} \frac{c o s (x^{2}) + s i n (x^{2})}{{(x^{2} + a^{2})}^{2}} d x w i t h a > 0

1) c a l c u l a t e f (a)

2) f i n d t h e v a l u e s o f \int_{0}^{\infty} \frac{c o s (x^{2}) + s i n (x^{2})}{{(x^{2} + 1)}^{2}} d x a n d

\int_{0}^{\infty} \frac{c o s (x^{2}) + s i n (x^{2})}{{(x^{2} + 3)}^{2}} d x

Commented by ~ À ® @ 237 ~ last updated on 23/Jul/19

w e a l w a y s h a v e c o s (x^{2}) + s i n (x^{2}) = 1

s o f (a) = \int_{0}^{\infty} \frac{1}{{(x^{2} + a^{2})}^{2}} d x

l e t c h a n g e x = a . t a n t d x = a (1 + {t a n}^{2} t) d t

f (a) = \int_{0}^{\frac{π}{2}} \frac{a (1 + {t a n}^{2} t) d t}{{(a^{2} {t a n}^{2} t + a^{2})}^{2}}

= \frac{1}{a^{3}} \int_{0}^{\frac{π}{2}} \frac{1}{(1 + {t a n}^{2} t)} d t

= \frac{1}{a^{3}} \int_{0}^{\frac{π}{2}} {c o s}^{2} t d t

k n o w i n g t h a t {c o s}^{2} t = \frac{1 + c o s 2 t}{2} w e f i n a l l y g o t

f (a) = \frac{1}{a^{3}} {[\frac{t}{2} + \frac{1}{4} s i n 2 t]}_{0}^{\frac{π}{2}}

= \frac{π}{4 a^{3}}

t h e n f (1) = \frac{π}{4} a n d f (\sqrt{3}) = \frac{π}{12 \sqrt{3}}

Commented by mathmax by abdo last updated on 23/Jul/19

t h a n k y o u s i r .

Commented by mathmax by abdo last updated on 23/Jul/19

r e a l l y i t s c o s (x^{2}) - s i n (x^{2}) n o t + b u t n e v e r m i n d i w i l l p o s t

a n o t h e r q u e s t i o n \dots

Commented by MJS last updated on 24/Jul/19

{(\cos x)}^{2} + {(\sin x)}^{2} = 1

but

\cos (x^{2}) + \sin (x^{2}) = \sqrt{2} \sin (x^{2} + \frac{π}{4})

which is n o t always = 1

Commented by mathmax by abdo last updated on 24/Jul/19

s i r \sim 237 y o u a n s w e r i s n o t c o r r e c t \dots .

Commented by mathmax by abdo last updated on 24/Jul/19

y o u a r e r i g h t s i r i h a v e c o m m i t e d a e r r o r i d e l e t t h i s p o s t

a n d g i v e t h e r i g h t a n s w e r \dots

Commented by mathmax by abdo last updated on 24/Jul/19

1) w e h a v e c o s (x^{2}) + s i n (x^{2}) = \sqrt{2} c o s (x^{2} - \frac{π}{4}) \Rightarrow

f (a) = \sqrt{2} \int_{0}^{\infty} \frac{c o s (x^{2} - \frac{π}{4})}{{(x^{2} + a^{2})}^{2}} d x \Rightarrow 2 f (a) = \sqrt{2} \int_{- \infty}^{+ \infty} \frac{c o s (x^{2} - \frac{π}{4})}{{(x^{2} + a^{2})}^{2}} d x

\Rightarrow \sqrt{2} f (a) = R e (\int_{- \infty}^{+ \infty} \frac{e^{i (x^{2} - \frac{π}{4})}}{{(x^{2} + a^{2})}^{2}} e x) l e t φ (z) = \frac{e^{i (z^{2} - \frac{π}{4})}}{{(z^{2} + a^{2})}^{2}} \Rightarrow

φ (z) = \frac{e^{i (z^{2} - \frac{π}{4})}}{{(z - i a)}^{2} {(z + i a)}^{2}} t h e p o l e s o f φ a r e \bar{+} i a (a > 0) 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, i a)

R e s (φ, i a) = {l i m}_{z \to i a} {(z - i a)}^{2} {{(z - i a)}^{2} φ (z)}^{(1)}

= {l i m}_{z \to i a} {\frac{e^{i (z^{2} - \frac{π}{4})}}{{(z + i a)}^{2}}}^{(1)} = e^{- \frac{i π}{4}} {l i m}_{z \to i a} {\frac{e^{{i z}^{2}}}{{(z + i a)}^{2}}}^{(1)}

= e^{- \frac{i π}{4}} {l i m}_{z \to i a} \frac{2 i z e^{{i z}^{2}} {(z + i a)}^{2} - 2 (z + i a) e^{{i z}^{2}}}{{(z + i a)}^{4}}

= e^{- \frac{i π}{4}} {l i m}_{z \to i a} \frac{(2 i z (z + i a) - 2) e^{{i z}^{2}}}{{(z + i a)}^{3}}

= e^{- i \frac{π}{4}} \frac{2 i (i a) (2 i a) - 2}{{(2 i a)}^{3}} e^{i {(i a)}^{2}} = e^{- \frac{i π}{4}} \frac{- 4 {i a}^{2} - 2}{- 8 {i a}^{3}} e^{- {i a}^{2}}

= \frac{2 {i a}^{2} - 1}{4 {i a}^{3}} e^{- i (\frac{π}{4} + a^{2})} \Rightarrow \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{2 {i a}^{2} - 1}{4 {i a}^{3}} e^{- i (\frac{π}{4} + a^{2})}

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

= - \frac{π}{2 a^{3}} {c o s (\frac{π}{4} + a^{2}) - i s i n (\frac{π}{4} + a^{2}) - 2 {i a}^{2} c o s (\frac{π}{4} + a^{2}) - 2 a^{2} s i n (\frac{π}{4} + a^{2})}

\sqrt{2} f (a) = - \frac{π}{2 a^{3}} (c o s (\frac{π}{4} + a^{2}) - 2 a^{2} s i n (\frac{π}{4} + a^{2}))

= - \frac{π}{2 a^{3}} c o s (\frac{π}{4} + a^{2}) + \frac{π}{a} s i n (\frac{π}{4} + a^{2}) \Rightarrow

f (a) = \frac{π}{a \sqrt{2}} s i n (\frac{π}{4} + a^{2}) - \frac{π}{2 \sqrt{2} a^{3}} c o s (\frac{π}{4} + a^{2})

Commented by mathmax by abdo last updated on 24/Jul/19

2) \int_{0}^{\infty} \frac{c o s (x^{2}) + s i n (x^{2})}{{(x^{2} + 1)}^{2}} d x = f (1) = \frac{π}{\sqrt{2}} s i n (\frac{π}{4} + 1) - \frac{π}{2 \sqrt{2}} c o s (\frac{π}{4} + 1)

\int_{0}^{\infty} \frac{c o s (x^{2}) + s i n (x^{2})}{{(x^{2} + 3)}^{2}} = f (\sqrt{3}) = \frac{π}{\sqrt{6}} s i n (\frac{π}{4} + 3) - \frac{π}{2 \sqrt{2} 3 \sqrt{3}} c o s (\frac{π}{4} + 3)

= \frac{π}{\sqrt{6}} s i n (3 + \frac{π}{4}) - \frac{π}{6 \sqrt{6}} c o s (3 + \frac{π}{4}) .

Commented by ~ À ® @ 237 ~ last updated on 25/Jul/19

y e s y o u a r e r i g h t . I d i d n o t m i n d t h e d i f f e r e n c e a t t h i s t i m e . S o r r y