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IntegrationQuestion and Answers: Page 243

Question Number 56345 Answers: 0 Comments: 1

let f(a) =∫_0 ^∞ (dx/(x^n +a^n )) with n integr ≥2 and a>0 1) calculate f(a) intems of a 2) let g(a) =∫_0 ^∞ (dx/((x^n +a^n )^2 )) calculate g(a) interms of a 3) find the values of integrals ∫_0 ^∞ (dx/(x^8 +16)) and ∫_0 ^∞ (dx/((x^8 +16)^2 ))

$l e t f (a) = \int_{0}^{\infty} \frac{d x}{x^{n} + a^{n}} w i t h n i n t e g r ⩾ 2 a n d a > 0$ $1) c a l c u l a t e f (a) i n t e m s o f a$ $2) l e t g (a) = \int_{0}^{\infty} \frac{d x}{{(x^{n} + a^{n})}^{2}} c a l c u l a t e g (a) i n t e r m s o f a$ $3) f i n d t h e v a l u e s o f i n t e g r a l s \int_{0}^{\infty} \frac{d x}{x^{8} + 16} a n d \int_{0}^{\infty} \frac{d x}{{(x^{8} + 16)}^{2}}$

Question Number 56311 Answers: 0 Comments: 1

let f(x) =∫_0 ^∞ ((cos(xt))/(x^2 +t^2 )) dt with x>0 1) find f(x) 2) find the values of ∫_0 ^∞ ((cos(t))/(1+t^2 ))dt and ∫_0 ^∞ ((cos(2t))/(4+t^2 ))dt 3) let U_n =∫_0 ^∞ ((cos(nt))/(n^2 +t^2 ))dt find lim_(n→+∞) U_n and study the convergenge of Σ U_n and Σ U_n ^2

$l e t f (x) = \int_{0}^{\infty} \frac{c o s (x t)}{x^{2} + t^{2}} d t w i t h x > 0$ $1) f i n d f (x)$ $2) f i n d t h e v a l u e s o f \int_{0}^{\infty} \frac{c o s (t)}{1 + t^{2}} d t a n d \int_{0}^{\infty} \frac{c o s (2 t)}{4 + t^{2}} d t$ $3) l e t U_{n} = \int_{0}^{\infty} \frac{c o s (n t)}{n^{2} + t^{2}} d t f i n d {l i m}_{n \to + \infty} U_{n} a n d s t u d y t h e c o n v e r g e n g e o f$ $Σ U_{n} a n d Σ U_{n}^{2}$

Question Number 56310 Answers: 0 Comments: 2

let f(x)=∫_(−∞) ^(+∞) cos(t^2 +xt +3)dt with x>0 1) find f(x) 2) calculate ∫_1 ^4 f(x)dx and ∫_1 ^(+∞) f(x)dx

$l e t f (x) = \int_{- \infty}^{+ \infty} c o s (t^{2} + x t + 3) d t w i t h x > 0$ $1) f i n d f (x)$ $2) c a l c u l a t e \int_{1}^{4} f (x) d x a n d \int_{1}^{+ \infty} f (x) d x$

Question Number 56280 Answers: 2 Comments: 2

Evaluate : 1) ((∫_0 ^( 1_ ) (1−(1−x^2 )^(100) )^(201) .xdx)/(∫_0 ^( 1) (1−(1−x^2 )^(100) )^(202) .xdx)) = ? 2) ((∫_0 ^( 1) (1−x^(200) )^(201) dx)/(∫_0 ^( 1) (1−x^(200) )^(202) dx)) = ?

$E v a l u a t e :$ $1) \frac{\int_{0}^{1_{}} {(1 - {(1 - x^{2})}^{100})}^{201} . x d x}{\int_{0}^{1} {(1 - {(1 - x^{2})}^{100})}^{202} . x d x} = ?$ $2) \frac{\int_{0}^{1} {(1 - x^{200})}^{201} d x}{\int_{0}^{1} {(1 - x^{200})}^{202} d x} = ?$

Question Number 56189 Answers: 0 Comments: 2

let u_n =∫_(−∞) ^∞ ((sin(nx^2 ))/(x^2 +x +n)) dx 1) calculate u_n 2) find lim_(n→+∞) u_n 3) study the serie Σ u_n

$l e t u_{n} = \int_{- \infty}^{\infty} \frac{s i n ({n x}^{2})}{x^{2} + x + n} d x$ $1) c a l c u l a t e u_{n}$ $2) f i n d {l i m}_{n \to + \infty} u_{n}$ $3) s t u d y t h e s e r i e Σ u_{n}$

Question Number 56188 Answers: 1 Comments: 0

find the value of ∫_0 ^∞ (((1+x)^(−(1/4)) −(1+x)^(−(3/4)) )/x) dx

$f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{{(1 + x)}^{- \frac{1}{4}} - {(1 + x)}^{- \frac{3}{4}}}{x} d x$

Question Number 56187 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ (((1+x)^α −(1+x)^β )/x) dx .

$s t u d y t h e c o n v e r g e n c e o f \int_{0}^{\infty} \frac{{(1 + x)}^{α} - {(1 + x)}^{β}}{x} d x .$

Question Number 56186 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(ix))/(2+x^2 ))dx

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

Question Number 56169 Answers: 1 Comments: 0

∫^1 _(−∞) (a+bi)^x dx=?

${\int_{- \infty}}^{1} {(a + b i)}^{x} d x = ?$

Question Number 56107 Answers: 2 Comments: 1

∫_(−1) ^0 ∣x sin (πx)∣ dx

$\int_{- 1}^{0} ∣ x \sin (π x) ∣ d x$

Question Number 56061 Answers: 1 Comments: 0

∫_0 ^1 e^(−x^2 ) dx correct to 3 decimal place.

$\int_{0}^{1} e^{- x^{2}} d x c o r r e c t t o 3 d e c i m a l p l a c e .$

Question Number 56060 Answers: 0 Comments: 3

∫e^(−x^2 ) dx as an infinite series.Hence investigate its converge.

$\int e^{- x^{2}} d x a s a n i n f i n i t e s e r i e s . H e n c e$ $i n v e s t i g a t e i t s c o n v e r g e .$

Question Number 55999 Answers: 0 Comments: 1

Question Number 55998 Answers: 1 Comments: 1

find f(x) =∫_0 ^1 arctan(t^2 +xt +1)dt .

$f i n d f (x) = \int_{0}^{1} a r c t a n (t^{2} + x t + 1) d t .$

Question Number 55997 Answers: 0 Comments: 0

calculate ∫_(π/3) ^(π/2) (dx/(x+sinx))

$c a l c u l a t e \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{d x}{x + s i n x}$

Question Number 55996 Answers: 0 Comments: 1

find ∫_(π/3) ^(π/2) (x/(cosx))dx

$f i n d \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{x}{c o s x} d x$

Question Number 55995 Answers: 1 Comments: 1

find I =∫ arctan(1−x)dx and J =∫ actan(1+x) dx

$f i n d I = \int a r c t a n (1 - x) d x a n d J = \int a c t a n (1 + x) d x$

Question Number 55994 Answers: 1 Comments: 0

calculate ∫_0 ^1 arctan(x^2 −x)dx

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

Question Number 55987 Answers: 0 Comments: 0

Question Number 55930 Answers: 2 Comments: 0

∫_0 ^( π) ((xtan x)/(sec x+tan x))dx = (is it (π^2 /2)−π)?

$\int_{0}^{π} \frac{x \tan x}{\sec x + \tan x} d x = (i s i t \frac{π^{2}}{2} - π) ?$

Question Number 55873 Answers: 2 Comments: 1

Integrate..∫(√(1+(√(1+(√x))))) dx

$I n t e g r a t e . . \int \sqrt{1 + \sqrt{1 + \sqrt{x}}} d x$

Question Number 55855 Answers: 0 Comments: 1

How to integrate ∫_0 ^1 ((sec^2 x)/(x(√x))) dx ?

$How to integrate \int_{0}^{1} \frac{\sec^{2} x}{x \sqrt{x}} d x ?$

Question Number 55834 Answers: 1 Comments: 0

Question Number 55760 Answers: 0 Comments: 3

let f(x) =∫_0 ^∞ ((cos(xt))/((xt^2 +i)^2 ))dx with x from R and x≠0 1) find a explicit form of f(x) 2) extract A =Re(f(x)) and B =Im(f(x)) and find its values . 3) calculate ∫_0 ^∞ ((cos(2t))/((2t^2 +i)^2 ))dt 4) let U_n =∫_0 ^∞ ((cos(nt))/((nt^2 +i)^2 ))dt .calculate lim_(n→+∞) u_n and study the convergence of Σu_n

$l e t f (x) = \int_{0}^{\infty} \frac{c o s (x t)}{{({x t}^{2} + i)}^{2}} d x w i t h x f r o m R a n d x \neq 0$ $1) f i n d a e x p l i c i t f o r m o f f (x)$ $2) e x t r a c t A = R e (f (x)) a n d B = I m (f (x)) a n d f i n d i t s v a l u e s .$ $3) c a l c u l a t e \int_{0}^{\infty} \frac{c o s (2 t)}{{(2 t^{2} + i)}^{2}} d t$ $4) l e t U_{n} = \int_{0}^{\infty} \frac{c o s (n t)}{{({n t}^{2} + i)}^{2}} d t . c a l c u l a t e {l i m}_{n \to + \infty} u_{n}$ $a n d s t u d y t h e c o n v e r g e n c e o f Σ u_{n}$

Question Number 55759 Answers: 1 Comments: 0

calculate I =∫_0 ^(2π) ((cost)/(3 +sin(2t)))dt and J =∫_0 ^(2π) ((sint)/(3 +cos(2t)))dt .

$c a l c u l a t e I = \int_{0}^{2 π} \frac{c o s t}{3 + s i n (2 t)} d t a n d J = \int_{0}^{2 π} \frac{s i n t}{3 + c o s (2 t)} d t .$

Question Number 55702 Answers: 0 Comments: 5

s=∫_0 ^( x) (√(1+(3t^2 +p)^2 ))dt = ? take p=1 for a special case.

$s = \int_{0}^{x} \sqrt{1 + {(3 t^{2} + p)}^{2}} d t = ?$ $t a k e p = 1 f o r a s p e c i a l c a s e .$

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