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IntegrationQuestion and Answers: Page 293

Question Number 34266 Answers: 0 Comments: 0

1) find the relation between ∫_x ^(+∞) e^(−t^2 ) dt and ∫_x ^(+∞) (e^(−t^2 ) /t^2 )dt 2) guive a equivalent to ∫_x ^(+∞) e^(−t^2 ) dt when x→+∞

$1) f i n d t h e r e l a t i o n b e t w e e n \int_{x}^{+ \infty} e^{- t^{2}} d t a n d$ $\int_{x}^{+ \infty} \frac{e^{- t^{2}}}{t^{2}} d t$ $2) g u i v e a e q u i v a l e n t t o \int_{x}^{+ \infty} e^{- t^{2}} d t w h e n x \to + \infty$

Question Number 34265 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ e^(−2t) sin([t]) dt .

$f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- 2 t} s i n ([t]) d t .$

Question Number 34264 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ e^(−2[t]) sint dt

$f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- 2 [t]} s i n t d t$

Question Number 34262 Answers: 0 Comments: 0

find the nature of ∫_2 ^(+∞) ((√(1+t^2 +t^4 )) −t ^3 (√(t^3 +at)))dt a∈R .

$f i n d t h e n a t u r e o f \int_{2}^{+ \infty} (\sqrt{1 + t^{2} + t^{4}} - t^{3} \sqrt{t^{3} + a t}) d t$ $a \in R .$

Question Number 34261 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ ((t−sint)/t^a )dt with a real.

$s t u d y t h e c o n v e r g e n c e o f \int_{0}^{\infty} \frac{t - s i n t}{t^{a}} d t w i t h a r e a l .$

Question Number 34260 Answers: 0 Comments: 0

let give a>0 1) find the value of F(a) = ∫_0 ^∞ ((lnt)/(t^2 +a^2 ))dt 2) find the value of G(a)=∫_0 ^∞ ((aln(t))/((t^2 +a^2 )^2 ))dt 3) find the value of ∫_0 ^∞ ((ln(t))/((t^2 +3)^2 ))dt

$l e t g i v e a > 0$ $1) f i n d t h e v a l u e o f F (a) = \int_{0}^{\infty} \frac{l n t}{t^{2} + a^{2}} d t$ $2) f i n d t h e v a l u e o f G (a) = \int_{0}^{\infty} \frac{a l n (t)}{{(t^{2} + a^{2})}^{2}} d t$ $3) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{l n (t)}{{(t^{2} + 3)}^{2}} d t$

Question Number 34257 Answers: 0 Comments: 0

find f(x)= ∫_1 ^x (dt/(t(√(1+t^2 )))) 2) calculate I =∫_1 ^(+∞) (dt/(t(√(1+t^2 ))))

$f i n d f (x) = \int_{1}^{x} \frac{d t}{t \sqrt{1 + t^{2}}}$ $2) c a l c u l a t e I = \int_{1}^{+ \infty} \frac{d t}{t \sqrt{1 + t^{2}}}$

Question Number 34255 Answers: 0 Comments: 0

find g(x)= ∫_0 ^∞ ((ln(1+xt^2 ))/t^2 ) dt 2) calculate ∫_0 ^∞ ((ln(1+3t^2 ))/t^2 )dt .

$f i n d g (x) = \int_{0}^{\infty} \frac{l n (1 + {x t}^{2})}{t^{2}} d t$ $2) c a l c u l a t e \int_{0}^{\infty} \frac{l n (1 + 3 t^{2})}{t^{2}} d t .$

Question Number 34254 Answers: 0 Comments: 0

find I(x)= ∫_0 ^1 ((ln(1+xt^2 ))/t^2 )dt .

$f i n d I (x) = \int_{0}^{1} \frac{l n (1 + {x t}^{2})}{t^{2}} d t .$

Question Number 34253 Answers: 0 Comments: 0

let F(x)= ∫_0 ^x ((ln(1+t^2 ))/t^2 )dt 1) calculate F(x) 2) find the value of ∫_0 ^∞ ((ln(1+t^2 ))/t^2 )dt

$l e t F (x) = \int_{0}^{x} \frac{l n (1 + t^{2})}{t^{2}} d t$ $1) c a l c u l a t e F (x)$ $2) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{l n (1 + t^{2})}{t^{2}} d t$

Question Number 34237 Answers: 0 Comments: 4

find ∫ (dx/(x^2 −a)) with a ∈ C .

$f i n d \int \frac{d x}{x^{2} - a} w i t h a \in C .$

Question Number 34229 Answers: 2 Comments: 3

calculate ∫_(−∞) ^∞ ((cos(tx))/(1+x^4 )) dx with t≥0 2) calculate ∫_0 ^∞ (dx/(1+x^4 )) .

$c a l c u l a t e \int_{- \infty}^{\infty} \frac{c o s (t x)}{1 + x^{4}} d x w i t h t ⩾ 0$ $2) c a l c u l a t e \int_{0}^{\infty} \frac{d x}{1 + x^{4}} .$

Question Number 34228 Answers: 0 Comments: 1

find the value of ∫_0 ^1 (x^2 /(1+x^4 ))dx

$f i n d t h e v a l u e o f \int_{0}^{1} \frac{x^{2}}{1 + x^{4}} d x$

Question Number 34227 Answers: 1 Comments: 2

calculate ∫_0 ^1 arctan(x^2 )dx

$c a l c u l a t e \int_{0}^{1} a r c t a n (x^{2}) d x$

Question Number 34225 Answers: 1 Comments: 0

find ∫ (dx/(1+x^2 +x^4 ))

$f i n d \int \frac{d x}{1 + x^{2} + x^{4}}$

Question Number 34223 Answers: 0 Comments: 0

find ∫ (dx/(x^(2n) −1)) with n integr natural and n≥1 .

$f i n d \int \frac{d x}{x^{2 n} - 1} w i t h n i n t e g r n a t u r a l a n d n ⩾ 1 .$

Question Number 34222 Answers: 0 Comments: 4

let give the sequence of integrals J_n =∫_0 ^∞ x^n e^(−(x^2 /2)) dx 1) prove that J_n =(n−1)J_(n−2) ∀n≥2 2) calculate J_(2p) and J_(2p+1) by using factoriels. 3) prove that ∀n≥1 J_n ^2 ≤J_(n−1) . J_(n+1) . 4)prove that ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p+1))) ≤J_0 ≤ ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p))) 5) find a equivalent of ((2^(2p) (p!)^2 )/((2p)!)) (p→+∞)

$l e t g i v e t h e s e q u e n c e o f i n t e g r a l s$ $J_{n} = \int_{0}^{\infty} x^{n} e^{- \frac{x^{2}}{2}} d x$ $1) p r o v e t h a t J_{n} = (n - 1) J_{n - 2} \forall n ⩾ 2$ $2) c a l c u l a t e J_{2 p} a n d J_{2 p + 1} b y u s i n g f a c t o r i e l s .$ $3) p r o v e t h a t \forall n ⩾ 1 J_{n}^{2} ⩽ J_{n - 1} . J_{n + 1} .$ $4) p r o v e t h a t \frac{2^{2 p} {(p!)}^{2}}{(2 p)!} \frac{1}{\sqrt{2 p + 1}} ⩽ J_{0} ⩽ \frac{2^{2 p} {(p!)}^{2}}{(2 p)!} \frac{1}{\sqrt{2 p}}$ $5) f i n d a e q u i v a l e n t o f \frac{2^{2 p} {(p!)}^{2}}{(2 p)!} (p \to + \infty)$