1-calculate-I-n-0-x-n-e-1-i-x-dx-with-n-integr-natural-and-i-2-1-2-find-0-x-4k-3-xsinx-dx-

Question Number 46612 by maxmathsup by imad last updated on 29/Oct/18

1) c a l c u l a t e I_{n} = \int_{0}^{\infty} x^{n} e^{(1 - i) x} d x w i t h n i n t e g r n a t u r a l a n d i^{2} = - 1

2) f i n d \int_{0}^{\infty} x^{4 k + 3} x s i n x d x .

Commented by maxmathsup by imad last updated on 29/Oct/18

2) t h e Q i s f i n d \int_{0}^{\infty} e^{- x} x^{4 k + 3} s i n x d x .

Commented by maxmathsup by imad last updated on 29/Oct/18

1) t h e Q i s {c a l c u l a t e I}_{n} = \int_{0}^{\infty} x^{n} e^{(- 1 + i) x} d x

Commented by maxmathsup by imad last updated on 29/Oct/18

1) w e h a v e I_{n} = \int_{0}^{\infty} x^{n} e^{(- 1 + i) x} d x = \int_{0}^{\infty} x^{n} e^{- (1 - i) x} d x c h a n g e m e n t (1 - i) x = t

g i v e I_{n} = \int_{0}^{\infty} {(\frac{t}{1 - i})}^{n} e^{- t} \frac{d t}{1 - i} = \frac{1}{{(1 - i)}^{n + 1}} \int_{0}^{\infty} t^{n} e^{- t} d t

= \frac{1}{{(1 - i)}^{n + 1}} Γ (n + 1) = \frac{n!}{{(1 - i)}^{n + 1}} = \frac{n!}{{(\sqrt{2} e^{- \frac{i π}{4}})}^{n + 1}}

= \frac{n!}{2^{\frac{n + 1}{2}}} e^{i (\frac{n + 1}{4} π)} = \frac{n!}{2^{\frac{n + 1}{2}}} {c o s (\frac{(n + 1) π}{4}) + i s i n (\frac{(n + 1) π}{4})}

Commented by maxmathsup by imad last updated on 29/Oct/18

2) w e h a v e \int_{0}^{\infty} e^{- x} x^{4 k + 3} s i n x d x = I m (\int_{0}^{\infty} x^{4 k + 3} e^{(- 1 + i) x} d x)

= I m (I_{4 k + 3}) = \frac{(4 k + 3)!}{2^{2 k + 2}} s i n (\frac{(4 k + 4) π}{4}) = \frac{(4 k + 3)!}{2^{2 k + 2}} s i n ((k + 1) π) = 0

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Oct/18

(1 - i) x = - t [t = (i - 1) x]

d x = \frac{- d t}{(1 - i)}

\int_{0}^{\infty} \frac{t^{n}}{{(i - 1)}^{n}} \times e^{- t} \times \frac{- d t}{(1 - i)}

\frac{1}{{(i - 1)}^{n + 1}} \int_{0}^{\infty} e^{- t} t^{n + 1 - 1} d t

\frac{1}{{(i - 1)}^{n + 1}} \times ⌈ (n + 1)

\frac{1}{{(i - 1)}^{n + 1}} \times n!

= \frac{{(- 1)}^{n + 1}}{{(1 - i)}^{n + 1}} \times \frac{{(1 + i)}^{n + 1}}{{(1 + i)}^{n + 1}} \times n!

= \frac{{(- 1)}^{n + 1}}{{(2)}^{n + 1}} \times {\sqrt{2} (\frac{1}{\sqrt{2}} + i \times \frac{1}{\sqrt{2}})}^{n + 1} \times n!

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

= \frac{{(- 1)}^{n + 1}}{2^{\frac{n + 1}{2}}} \times {c o s (2 k π + \frac{π}{4}) + i s i n (2 k π + \frac{π}{4})}^{n + 1} \times n!