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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 62341 Answers: 2 Comments: 3

How many real root does the equation x^8 − x^7 + 2x^6 − 2x^5 + 3x^4 − 3x^3 + 4x^2 − 4x + (5/2) = 0 has

$How many real root does the equation$ $x^{8} - x^{7} + {2 x}^{6} - {2 x}^{5} + {3 x}^{4} - {3 x}^{3} + {4 x}^{2} - 4 x + \frac{5}{2} = 0 has$

Question Number 62340 Answers: 0 Comments: 3

Question Number 62338 Answers: 0 Comments: 0

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 62334 Answers: 2 Comments: 0

if α^2 +β^2 = (α+β)^2 −2αβ evaluate(α−β)

$i f α^{2} + β^{2} = {(α + β)}^{2} - 2 α β e v a l u a t e (α - β)$

Question Number 62332 Answers: 1 Comments: 0

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 62322 Answers: 0 Comments: 2

Question Number 62308 Answers: 1 Comments: 2

Question Number 62291 Answers: 1 Comments: 1

Question Number 62288 Answers: 0 Comments: 0

M_(TP) =Q(D/Z) × f_

$M_{T P} = Q \frac{D}{Z} \times f_{}$

Question Number 62289 Answers: 3 Comments: 0

Question Number 62281 Answers: 1 Comments: 0

{ ((x^3 +y^3 =3xy)),((x^4 +y^4 =4xy)) :} [x,y≠0]

${\begin{cases} x^{3} + y^{3} = 3 xy \\ x^{4} + y^{4} = 4 xy \end{cases} [x, y \neq 0]$

Question Number 62276 Answers: 1 Comments: 0

{ ((((√x)/a)+((√y)/b)=1)),((((√a)/x)+((√b)/y)=1)) :} a,b∈R^+

${\begin{cases} \frac{\sqrt{x}}{a} + \frac{\sqrt{y}}{b} = 1 \\ \frac{\sqrt{a}}{x} + \frac{\sqrt{b}}{y} = 1 \end{cases} a, b \in R^{+}$

Question Number 62275 Answers: 1 Comments: 0

{ ((a(√x)+b(√y)=2(√(ab)))),((x(√a)+y(√b)=2(√(ab)))) :} a,b∈R^+

${\begin{cases} a \sqrt{x} + b \sqrt{y} = 2 \sqrt{ab} \\ x \sqrt{a} + y \sqrt{b} = 2 \sqrt{ab} \end{cases} a, b \in R^{+}$

Question Number 62274 Answers: 1 Comments: 4

∫_0 ^∞ e^(−x^2 ) dx

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

Question Number 62273 Answers: 0 Comments: 4

Mr. Rasheed.Sindhi I sense you′re much engaged in making olympiad contents these days , I wish that you join my workspace concerning that same.

$M r . R a s h e e d . S i n d h i$ $I s e n s e {y o u}^{'} r e m u c h e n g a g e d i n m a k i n g$ $o l y m p i a d c o n t e n t s t h e s e d a y s, I w i s h$ $t h a t y o u j o i n m y w o r k s p a c e c o n c e r n i n g t h a t s a m e .$

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 62265 Answers: 1 Comments: 1

Question Number 62263 Answers: 1 Comments: 0

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}

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

Question Number 62251 Answers: 0 Comments: 1

∫(x^2 −4)^(1/2) dx trig substitution only

$\int {(x^{2} - 4)}^{1 / 2} dx$ $trig substitution only$

Question Number 62244 Answers: 1 Comments: 3

Find out x,y, such that gcd(x^3 ,y^2 )=gcd(x^2 ,y^3 )

$Find out x, y, such that$ $\gcd (x^{3}, y^{2}) = \gcd (x^{2}, y^{3})$

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