decompose-inside-C-x-the-fraction-F-x-1-x-2-1-n-calculate-0-F-x-dx-

Question Number 73484 by abdomathmax last updated on 13/Nov/19

d e c o m p o s e i n s i d e C (x) t h e f r a c t i o n

F (x) = \frac{1}{{(x^{2} + 1)}^{n}}

c a l c u l a t e \int_{0}^{\infty} F (x) d x

Answered by mind is power last updated on 13/Nov/19

F (x) = \frac{1}{2} \int_{- \infty}^{+ \infty} \frac{d x}{{(x^{2} + 1)}^{n}}

= i π R e s (\frac{1}{{(x^{2} + 1)}^{n}}, i)

= i π . \frac{1}{(n - 1)!} \frac{d^{n - 1}}{{d x}^{n - 1}} . {(x - i)}^{n} . \frac{1}{{(x - i)}^{n} {(x + i)}^{n}} \underset{x = i}{∣}

f (x) = \frac{1}{{(x + i)}^{n}}, f^{(k)} = \frac{{(- 1)}^{k} . n . (n + 1) \dots (n + k - 1)}{{(x + i)}^{n + k}}

f^{(n -)} (i) = \frac{{(- 1)}^{n - 1} . n \dots . . (2 n - 2)}{{(2 i)}^{2 n - 1}} = - 2 i \frac{n \dots . (2 n - 2)}{4^{n}} = - 2 i \frac{(2 n - 2)!}{4^{n} . (n - 1)!}

= - 2 i (n - 1)! . C_{2 n - 2}^{n - 1}

F (x) = \frac{1}{2} \int_{- \infty}^{+ \infty} \frac{d x}{{(x^{2} + 1)}^{n}} = . \frac{i π}{(n - 1)!} . - 2 i (n - 1) C_{2 n - 1}^{n - 1} . \frac{1}{4^{n}} = π \frac{C_{2 n - 2}^{n - 1}}{2^{2 n - 1}}

2 n d M e t h o d e

\int_{0}^{+ \infty} \frac{d x}{(x^{2} + 1)} = \int_{0}^{\frac{π}{2}} \frac{d (t g (t))}{{(1 + {t g}^{2} (t))}^{n}} = \int_{0}^{\frac{π}{2}} \frac{(1 + {t g}^{2} (t)) d t}{{(1 + {t g}^{2} (t))}^{n}}

= \int_{0}^{\frac{π}{2}} \frac{d t}{{(1 + {t g}^{2} (t))}^{n - 1}} = \int_{0}^{\frac{π}{2}} {c o s}^{2 n - 2} (t) d t = W_{2 n - 2} . . w a l i s e

o r, u s e B (x, y) = 2 \int_{0}^{\frac{π}{2}} {c o s}^{2 x - 1} (t) {s i n}^{2 y - 1} (t) d t

B (n - \frac{1}{2}, \frac{1}{2}) = 2 \int_{0}^{\frac{π}{2}} . {c o s}^{2 n - 2} (t) d t = \frac{Γ (n - \frac{1}{2}) . Γ (\frac{1}{2})}{Γ (n)}

Γ (\frac{1}{2}) = \sqrt{π}

Γ (n - \frac{1}{2}) = Γ (\frac{2 n - 1}{2}) = Γ (\frac{2 n - 3}{2} + 1) = \frac{(2 n - 3)}{2} \dots \dots \frac{1}{2} . \sqrt{π}

= \frac{(2 n - 3) \dots . . 1 .}{2^{n - 1}} \sqrt{π} \dots n ⩾ 2

B (n - \frac{1}{2}, \frac{1}{2}) = \frac{\sqrt{π}}{2^{n - 1}} . (2 n - 3) \dots . (1) . \sqrt{π} . \frac{1}{(n - 1)!}

= \frac{π . (2 n - 3) \dots . 1 \dots . . 2.4 \dots . . (2 n - 2)}{2^{n - 1} (n - 1)! . 2^{n - 1} . (n - 1)!} = π \frac{C_{2 n - 2}^{n - 1}}{2^{2 n - 2}} = 2 \int_{0}^{\frac{π}{2}} {c o s}^{2 n - 2} (t) d t

\Rightarrow \int_{0}^{\frac{π}{2}} {c o s}^{2 n - 2} (t) d t = \frac{π}{2} \frac{C_{2 n - 2}^{n - 1}}{2^{2 n - 2}} = π \frac{C_{2 n - 2}^{n - 1}}{2^{2 n - 1}}

Commented by abdomathmax last updated on 17/Nov/19

t h s n k x s i r .

Commented by mind is power last updated on 17/Nov/19

y^{'} re welcom