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IntegrationQuestion and Answers: Page 230

Question Number 62731 Answers: 0 Comments: 1

find ∫ (√((x−1)/(x^2 +3)))dx

$f i n d \int \sqrt{\frac{x - 1}{x^{2} + 3}} d x$

Question Number 62653 Answers: 1 Comments: 4

∫x(arctan(x))^2 dx ∫((x e^(arctan(x)) )/((1+x^2 )^(3/2) )) dx ∫((arcsin(x))/(√(1+x))) dx

$\int x {(\arctan (x))}^{2} dx$ $\int \frac{x e^{\arctan (x)}}{{(1 + x^{2})}^{\frac{3}{2}}} dx$ $\int \frac{\arcsin (x)}{\sqrt{1 + x}} dx$

Question Number 62648 Answers: 0 Comments: 1

Question Number 62613 Answers: 3 Comments: 2

Question Number 62596 Answers: 1 Comments: 0

∫sin^(100) (x) cos^(100) (x) dx

$\int \sin^{100} (x) \cos^{100} (x) dx$

Question Number 62455 Answers: 0 Comments: 3

Question Number 62453 Answers: 0 Comments: 3

∫ (x/(e^x − 1))dx, for x > 0

$\int \frac{x}{e^{x} - 1} dx, for x > 0$

Question Number 62437 Answers: 0 Comments: 1

let f(x) =∫_0 ^1 ((arctan(1+xt))/(t^2 +1))dt determine a explicit form for f(x) 2)calculate ∫_0 ^1 ((arctan(1+2t))/(1+t^2 ))dt

$l e t f (x) = \int_{0}^{1} \frac{a r c t a n (1 + x t)}{t^{2} + 1} d t$ $d e t e r m i n e a e x p l i c i t f o r m f o r f (x)$ $2) c a l c u l a t e \int_{0}^{1} \frac{a r c t a n (1 + 2 t)}{1 + t^{2}} d t$

Question Number 62425 Answers: 0 Comments: 0

let ξ(x) =Σ_(n=1) ^∞ (1/n^x ) with x>1 1) calculate lim_(x→1^+ ) ξ(x) and lim_(x→+∞) ξ(x) 2) prove that ξ(x) =1+2^(−x) +o(2^(−x) ) (x→+∞) 3) prove that ξ is decreasing and convexe fucntion on]1,+∞[

$l e t ξ (x) = \sum_{n = 1}^{\infty} \frac{1}{n^{x}} w i t h x > 1$ $1) c a l c u l a t e {l i m}_{x \to 1^{+}} ξ (x) a n d {l i m}_{x \to + \infty} ξ (x)$ $2) p r o v e t h a t ξ (x) = 1 + 2^{- x} + o (2^{- x}) (x \to + \infty)$ $3) p r o v e t h a t ξ i s d e c r e a s i n g a n d c o n v e x e f u c n t i o n o n] 1, + \infty [$

Question Number 62420 Answers: 0 Comments: 0

let u_n (x)=(1/n^x ) −∫_n ^(n+1) (dt/t^x ) with x∈[1,2] 1)prove that 0≤ u_n (x)≤(1/n^x )−(1/((n+1)^x )) (n>0) 2)prove that Σ u_n (x)converges let γ =Σ_(n=1) ^∞ u_n (1) 3)find Σ_(n=1) ^∞ u_n (x) interms of ξ(x)and 1−x 4) prove that the converg.of Σu_n (x)is uniform prove that for x∈V(1) ξ(x) =(1/(x−1)) +γ +o(1) 5) find the value of Σ_(n=1) ^∞ (((−1)^(n−1) )/n)ln(n)

$l e t u_{n} (x) = \frac{1}{n^{x}} - \int_{n}^{n + 1} \frac{d t}{t^{x}} w i t h x \in [1, 2]$ $1) p r o v e t h a t 0 ⩽ u_{n} (x) ⩽ \frac{1}{n^{x}} - \frac{1}{{(n + 1)}^{x}} (n > 0)$ $2) p r o v e t h a t Σ u_{n} (x) c o n v e r g e s$ $l e t γ = \sum_{n = 1}^{\infty} u_{n} (1)$ $3) f i n d \sum_{n = 1}^{\infty} u_{n} (x) i n t e r m s o f ξ (x) a n d$ $1 - x$ $4) p r o v e t h a t t h e c o n v e r g . o f Σ u_{n} (x) i s$ $u n i f o r m$ $p r o v e t h a t f o r x \in V (1)$ $ξ (x) = \frac{1}{x - 1} + γ + o (1)$ $5) f i n d t h e v a l u e o f \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n} l n (n)$

Question Number 62419 Answers: 0 Comments: 1

calculate ∫_0 ^1 (2x^2 −1)(√(x^2 −2x+5))dx

$c a l c u l a t e \int_{0}^{1} (2 x^{2} - 1) \sqrt{x^{2} - 2 x + 5} d x$

Question Number 62418 Answers: 0 Comments: 0

calculate ∫_0 ^1 Γ(t).Γ(1−t)dt

$c a l c u l a t e \int_{0}^{1} Γ (t) . Γ (1 - t) d t$

Question Number 62417 Answers: 0 Comments: 0

prove that Γ(x).Γ(1−x) =(π/(sin(πx))) with 0<x<1

$p r o v e t h a t Γ (x) . Γ (1 - x) = \frac{π}{s i n (π x)} w i t h 0 < x < 1$

Question Number 62416 Answers: 0 Comments: 1

let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>1 calculate Γ^((n)) (x) for all integr n.

$l e t Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t w i t h x > 1 c a l c u l a t e Γ^{(n)} (x) f o r a l l i n t e g r n .$

Question Number 62415 Answers: 0 Comments: 1

calculate f(x,y) =∫_0 ^∞ e^(−xt) ln(yt) dt with x>0 and y>0 .

$c a l c u l a t e f (x, y) = \int_{0}^{\infty} e^{- x t} l n (y t) d t w i t h x > 0 a n d y > 0 .$

Question Number 62414 Answers: 1 Comments: 0

find ∫ (e^x /(√(e^(2x) −1)))dx

$f i n d \int \frac{e^{x}}{\sqrt{e^{2 x} - 1}} d x$

Question Number 62412 Answers: 0 Comments: 2

calculate lim_(n→+∞) ∫_0 ^n (1−(x/n))^n dx

$c a l c u l a t e {l i m}_{n \to + \infty} \int_{0}^{n} {(1 - \frac{x}{n})}^{n} d x$

Question Number 62389 Answers: 1 Comments: 1

∫0dx= help

$\int 0 dx =$ $help$

Question Number 62343 Answers: 1 Comments: 1

calculate ∫_0 ^(π/4) {xΠ_(k=1) ^∞ cos((x/2^k ))}dx

$c a l c u l a t e \int_{0}^{\frac{π}{4}} {x \prod_{k = 1}^{\infty} c o s (\frac{x}{2^{k}})} d x$

Question Number 62342 Answers: 1 Comments: 4

let f(ξ) =∫ (x^2 /(√(1−ξx^2 )))dx with 0<ξ<1 1) determine a explicit form of f(ξ) 2) calculate lim_(ξ→1) f(ξ) 3) calculate ∫_0 ^(1/2) (x^2 /(√(1−sin^2 θ x^2 ))) dx with 0<θ<(π/2)

$l e t f (ξ) = \int \frac{x^{2}}{\sqrt{1 - ξ x^{2}}} d x w i t h 0 < ξ < 1$ $1) d e t e r m i n e a e x p l i c i t f o r m o f f (ξ)$ $2) c a l c u l a t e {l i m}_{ξ \to 1} f (ξ)$ $3) c a l c u l a t e \int_{0}^{\frac{1}{2}} \frac{x^{2}}{\sqrt{1 - {s i n}^{2} θ x^{2}}} d x w i t h 0 < θ < \frac{π}{2}$

Question Number 62335 Answers: 0 Comments: 2

1) calculate f(x,y) =∫_0 ^∞ ((e^(−xt) cos(yt))/(√t)) dt and g(x,y) =∫_0 ^∞ ((e^(−xt) sin(yt))/(√t)) dt with x>0 and y>0 2) find the values of ∫_0 ^∞ ((e^(−2t) cos(t))/(√t)) dt and ∫_0 ^∞ ((e^(−t) cos(2t))/(√t)) dt

$1) c a l c u l a t e f (x, y) = \int_{0}^{\infty} \frac{e^{- x t} c o s (y t)}{\sqrt{t}} d t a n d g (x, y) = \int_{0}^{\infty} \frac{e^{- x t} s i n (y t)}{\sqrt{t}} d t$ $w i t h x > 0 a n d y > 0$ $2) f i n d t h e v a l u e s o f \int_{0}^{\infty} \frac{e^{- 2 t} c o s (t)}{\sqrt{t}} d t a n d \int_{0}^{\infty} \frac{e^{- t} c o s (2 t)}{\sqrt{t}} d t$

Question Number 62330 Answers: 1 Comments: 1

find the value of ∫_0 ^∞ (t^(a−1) /((1+t)^2 ))dt with 0<a<1

$f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{t^{a - 1}}{{(1 + t)}^{2}} d t w i t h 0 < a < 1$

Question Number 62274 Answers: 1 Comments: 4

∫_0 ^∞ e^(−x^2 ) dx

$\underset{0}{\int^{\infty}} e^{- x^{2}} d x$

Question Number 62266 Answers: 1 Comments: 1

∫((2sin(x)+3cos(x))/(3sin(x)+4cos(x)))dx

$\int \frac{2 s i n (x) + 3 c o s (x)}{3 s i n (x) + 4 c o s (x)} d x$

Question Number 62262 Answers: 1 Comments: 1

find the value of I =∫_0 ^∞ ((e^(−t) sint)/(√t))dt and J =∫_0 ^∞ ((e^(−t) cos(t))/(√t))dt ,study first the convergence.

$f i n d t h e v a l u e o f$ $I = \int_{0}^{\infty} \frac{e^{- t} s i n t}{\sqrt{t}} d t a n d J = \int_{0}^{\infty} \frac{e^{- t} c o s (t)}{\sqrt{t}} d t, s t u d y f i r s t t h e c o n v e r g e n c e .$

Question Number 62252 Answers: 0 Comments: 1

∫ln(x+1)/(x^2 −x+1) limit ={ 0>2}

$\int \ln (x + 1) / (x^{2} - x + 1)$ $limit = {0 > 2}$

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