let f_n (t) =∫_0 ^∞ (dx/((x^2 +t^2 )^n )) with n from N and n≥1 1. find a explicit form of f_n (t) 2. what is the value of g_n (t)=∫_0 ^∞ ((t dx)/((x^2 +t^2 )^(n+1) )) ? 3. calculate ∫_0 ^∞ (dx/((x^2 +3)^4 )) and ∫


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Question Number 56932 by turbo msup by abdo last updated on 27/Mar/19

$l e t f_{n} (t) = \int_{0}^{\infty} \frac{d x}{{(x^{2} + t^{2})}^{n}}$ $w i t h n f r o m N a n d n ⩾ 1$ $1 . f i n d a e x p l i c i t f o r m o f f_{n} (t)$ $2 . w h a t i s t h e v a l u e o f$ $g_{n} (t) = \int_{0}^{\infty} \frac{t d x}{{(x^{2} + t^{2})}^{n + 1}} ?$ $3 . c a l c u l a t e \int_{0}^{\infty} \frac{d x}{{(x^{2} + 3)}^{4}}$ $a n d \int_{0}^{\infty} \frac{d x}{{(x^{2} + 16)}^{3}}$
	Answered by Smail last updated on 27/Mar/19

	$l e t x = t \times t a n θ \Rightarrow d x = \frac{t d θ}{{c o s}^{2} θ}$ $f_{n} (t) = \int_{0}^{π / 2} \frac{t}{t^{2 n} {c o s}^{2} θ {(1 + {t a n}^{2} θ)}^{n}} d θ$ $= t^{1 - 2 n} \int_{0}^{π / 2} {c o s}^{2 (n - 1)} θ d θ$ $l e t A_{n - 1} = \int_{0}^{π / 2} {c o s}^{2 (n - 1)} θ d θ$ $= \int_{0}^{π / 2} ({c o s}^{2 (n - 2)} θ - {s i n}^{2} θ {c o s}^{2 (n - 2)} θ) d θ$ $B y p a r t s$ $u = s i n θ \Rightarrow u^{'} = c o s θ$ $v^{'} = s i n θ {c o s}^{2 n - 4} θ \Rightarrow v = - \frac{1}{2 n - 3} {c o s}^{2 n - 3}$ $A_{n - 1} = A_{n - 2} - \frac{1}{2 n - 3} A_{n - 1}$ $A_{n - 1} = \frac{2 n - 3}{2 n - 2} A_{n - 2} = \frac{(2 n - 3) (2 n - 5)}{(2 n - 2) (2 n - 4)} A_{n - 3}$ $= \frac{(2 (n - 1) - 1) (2 (n - 2) - 1) (2 (n - 3) - 1) . . . (2 (n - (n - 1)) - 1)}{(2 (n - 1)) (2 (n - 2)) . . . (2 (n - (n - 1))} A_{n - (n - 1)}$ $= \frac{(2 n - 3) (2 n - 5) (2 n - 7) . . . 1}{2^{n - 1} (n - 1)!} \int_{0}^{π / 2} {c o s}^{2} θ d θ$ $= \frac{(2 n - 2) (2 n - 3) (2 n - 4) (2 n - 5) (2 n - 6) . . . 1}{2^{n - 1} (n - 1)! \times 2^{n - 1} (n - 1)!} \times \frac{1}{2} {[θ + \frac{1}{2} s i n 2 θ]}_{0}^{π / 2}$ $= \frac{(2 n - 2)!}{2^{2 (n - 1)} {((n - 1)!)}^{2}} \times \frac{π}{2 \times 2}$ $A_{n - 1} = \frac{π (2 n - 2)!}{2^{2 n} {((n - 1)!)}^{2}}$ $f_{n} (t) = t^{1 - 2 n} \times \frac{π (2 n - 2)!}{2^{2 n} {((n - 1)!)}^{2}}$ $f_{n} (t) = \frac{π (2 n - 2)!}{2^{2 n} {((n - 1)!)}^{2}} t^{1 - 2 n}$
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