let-f-a-cos-ax-2-dx-with-a-gt-0-1-calculate-f-a-interms-of-a-calculate-cos-2x-2-dx-3-find-the-value-of-cos-x-2-x-1-dx-

Question Number 42260 by math khazana by abdo last updated on 21/Aug/18

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

1) c a l c u l a t e f (a) i n t e r m s o f a

) c a l c u l a t e \int_{- \infty}^{+ \infty} c o s (2 x^{2}) d x

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

Commented by maxmathsup by imad last updated on 21/Aug/18

3) \int_{- \infty}^{+ \infty} c o s (x^{2} + x + 1) d x = \int_{- \infty}^{+ \infty} c o s ({(x + \frac{1}{2})}^{2} + \frac{3}{4}) d x

=_{x + \frac{1}{2} = \frac{\sqrt{3}}{2} t} \int_{- \infty}^{+ \infty} c o s (\frac{3}{4} (t^{2} + 1)) \frac{\sqrt{3}}{2} d t

= \frac{\sqrt{3}}{2} \int_{- \infty}^{+ \infty} c o s (\frac{3}{4} t^{2} + \frac{3}{4}) d t

= \frac{\sqrt{3}}{2} \int_{- \infty}^{+ \infty} {c o s (\frac{3}{4} t^{2}) c o s (\frac{3}{4}) - s i n (\frac{3}{4} t^{2}) s i n (\frac{3}{4})} d t

= \frac{\sqrt{3}}{2} c o s (\frac{3}{4}) \int_{- \infty}^{+ \infty} c o s (\frac{3}{4} t^{2}) d t - \frac{\sqrt{3}}{2} s i n (\frac{3}{4}) \int_{- \infty}^{+ \infty} s i n (\frac{3}{4} t^{2}) d t

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

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

\int_{- \infty}^{+ \infty} c o s (x^{2} + x + 1) d x = \sqrt{\frac{π}{2}} {c o s (\frac{3}{4}) - s i n (\frac{3}{4})} .

Commented by maxmathsup by imad last updated on 21/Aug/18

w e h a v e \int_{- \infty}^{+ \infty} c o s ({a x}^{2}) d x - i \int_{- \infty}^{+ \infty} s i n ({a x}^{2}) d x

= \int_{- \infty}^{+ \infty} e^{- {i a x}^{2}} d x = \int_{- \infty}^{+ \infty} e^{- {(\sqrt{i a} x)}^{2}} d x =_{\sqrt{i a} x = t} \int_{- \infty}^{+ \infty} e^{- t^{2}} \frac{d t}{\sqrt{i a}}

= \frac{1}{\sqrt{a} e^{\frac{i π}{4}}} \sqrt{π} = \frac{\sqrt{π}}{\sqrt{a}} e^{- \frac{i π}{4}} = \frac{\sqrt{π}}{\sqrt{a}} {c o s (\frac{π}{4}) - i s i n (\frac{π}{4})} \Rightarrow

f (a) = \frac{\sqrt{π}}{\sqrt{a}} \frac{\sqrt{2}}{2} = \frac{\sqrt{2 π}}{2 \sqrt{a}} . a l s o w e h a v e \int_{- \infty}^{+ \infty} s i n ({a x}^{2}) d x = \frac{\sqrt{2 π}}{2 \sqrt{a}}

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