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

Question Number 65286 by mathmax by abdo last updated on 27/Jul/19

1) f i n d f (a) = \int_{- \infty}^{+ \infty} e^{- {a x}^{2}} c o s (3 - x^{2}) d x w i t h a > 0

2) f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- 3 x^{2}} c o s (3 - x^{2}) d x

Commented by mathmax by abdo last updated on 29/Jul/19

1) w e h a v e f (a) = \int_{- \infty}^{+ \infty} e^{- {a x}^{2}} c o s (3 - x^{2}) d x \Rightarrow

f (a) = R e (\int_{- \infty}^{+ \infty} e^{- {a x}^{2} + i (3 - x^{2})} d x)

I = \int_{- \infty}^{+ \infty} e^{- {a x}^{2} + i (3 - x^{2})} d x = e^{3 i} \int_{- \infty}^{+ \infty} e^{- (a + i) x^{2}} d x c h a n g e m e n t \sqrt{a + i} x = t

g i v e I = e^{3 i} \int_{- \infty}^{+ \infty} e^{- t^{2}} \frac{d t}{\sqrt{a + i}} = \frac{e^{3 i}}{\sqrt{a + i}} \sqrt{π}

a + i = \sqrt{1 + a^{2}} {\frac{a}{\sqrt{1 + a^{2}}} + \frac{i}{\sqrt{1 + a^{2}}}} = r e^{i θ} \Rightarrow r = \sqrt{1 + a^{2}} a n d θ = a r c t a n (\frac{1}{a})

\sqrt{a + i} = \sqrt{r} e^{\frac{i}{2} θ} = {(1 + a^{2})}^{\frac{1}{4}} e^{\frac{i}{2} a r c t a n (\frac{1}{a})} =^{4} \sqrt{1 + a^{2}} e^{\frac{i}{2} (\frac{π}{2} - a r c t a n a)}

=^{4} \sqrt{1 + a^{2}} e^{\frac{i π}{4} - \frac{i}{2} a r c t a n a} \Rightarrow

I = \sqrt{π} \frac{e^{3 i}}{(^{4} \sqrt{1 + a^{2}}) e^{\frac{i π}{4} - \frac{i}{2} a r c t a n (a)}} = \frac{\sqrt{π}}{(^{4} \sqrt{1 + a^{2})}} e^{3 i - \frac{i π}{4} + \frac{i}{2} a r c t a n (a)}

= \frac{\sqrt{π}}{(^{4} \sqrt{1 + a^{2}})} e^{i (3 - \frac{π}{4} + \frac{1}{2} a r c t a n (a))} \Rightarrow

f (a) = \frac{\sqrt{π}}{(^{4} \sqrt{1 + a^{2})}} c o s (3 - \frac{π}{4} + \frac{a r c t a n a}{2}) .

2) \int_{0}^{\infty} e^{- 3 x^{2}} c o s (3 - x^{2}) d x = \frac{1}{2} \int_{- \infty}^{+ \infty} e^{- 3 x^{2}} c o s (3 - x^{2}) d x

= \frac{1}{2} f (3) = \frac{\sqrt{π}}{(^{4} \sqrt{10})} c o s (3 - \frac{π}{4} + \frac{a r c t a n (3)}{2})

Commented by mathmax by abdo last updated on 29/Jul/19

\int_{0}^{\infty} e^{- 3 x^{2}} c o s (3 - x^{2}) d x = \frac{\sqrt{π}}{2 (^{4} \sqrt{10})} c o s (3 - \frac{π}{4} + \frac{a r c t a n (3)}{2})