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Question Number 139348 Answers: 0 Comments: 0

Question Number 139342 Answers: 2 Comments: 1

Question Number 139332 Answers: 1 Comments: 0

Find the coefficient of x^(50) in the expression (1+x)^(1000) +2x(1+x)^(999) + 3x^2 (1+x)^(998) +...+1001x^(1000)

$Find the coefficient of x^{50} in$ $the expression {(1 + x)}^{1000} + 2 x {(1 + x)}^{999} +$ ${3 x}^{2} {(1 + x)}^{998} + . . . + {1001 x}^{1000}$

Question Number 139331 Answers: 2 Comments: 0

(√x)+(√y)=3 and (√(x+5))+(√(y+3))=5 find x and y

$\sqrt{x} + \sqrt{y} = 3 a n d$ $\sqrt{x + 5} + \sqrt{y + 3} = 5$ $f i n d x a n d y$

Question Number 139328 Answers: 2 Comments: 1

Question Number 139323 Answers: 0 Comments: 0

Let f define such as f(1)=1,f(3)=3 ∀ n≥2 , f(2n)=f(n) f(4n+1)=2f(2n+1)−f(n) f(4n+3)=3f(2n+1)−2f(n) 1)Prove that ∀ n , f(n) is odd 2)Prove that if f(a_n )=a_n , then a_n =2^n −1 or a_n =2^n +1

$L e t f d e f i n e s u c h a s f (1) = 1, f (3) = 3$ $\forall n ⩾ 2, f (2 n) = f (n)$ $f (4 n + 1) = 2 f (2 n + 1) - f (n)$ $f (4 n + 3) = 3 f (2 n + 1) - 2 f (n)$ $1) P r o v e t h a t \forall n, f (n) i s o d d$ $2) P r o v e t h a t i f f (a_{n}) = a_{n},$ $t h e n a_{n} = 2^{n} - 1 o r a_{n} = 2^{n} + 1$

Question Number 139319 Answers: 0 Comments: 2

prove that are axactly 1729 positive integer solutions to the below equation 4x^4 +3y^3 +2z^2 +t=4311

$p r o v e t h a t a r e a x a c t l y 1729 p o s i t i v e$ $i n t e g e r s o l u t i o n s t o t h e b e l o w e q u a t i o n$ $4 x^{4} + 3 y^{3} + 2 z^{2} + t = 4311$

Question Number 139371 Answers: 3 Comments: 0

by use Gamma function prove (1) ∫_0 ^( (π/8)) cos^3 4xdx=(1/6) (2) ∫_0 ^( π) sin^6 ((x/2))cos^8 ((x/2))dx=((5π)/2^(11) )

$b y u s e G a m m a f u n c t i o n p r o v e$ $(1) \int_{0}^{\frac{π}{8}} {c o s}^{3} 4 x d x = \frac{1}{6}$ $(2) \int_{0}^{π} {s i n}^{6} (\frac{x}{2}) {c o s}^{8} (\frac{x}{2}) d x = \frac{5 π}{2^{11}}$

Question Number 139313 Answers: 2 Comments: 0

∫_0 ^∞ (dx/((1+x^5 )^5 ))=((798(√2))/(5^5 (√(5−(√5)))))π

$\int_{0}^{\infty} \frac{d x}{{(1 + x^{5})}^{5}} = \frac{798 \sqrt{2}}{5^{5} \sqrt{5 - \sqrt{5}}} π$

Question Number 139296 Answers: 4 Comments: 0

∫_0 ^( 1) ((√x)/( (√(1−x))+(√x)))dx

$\int_{0}^{1} \frac{\sqrt{x}}{\sqrt{1 - x} + \sqrt{x}} d x$

Question Number 139290 Answers: 2 Comments: 0

∫_( 0) ^( ∞) ((sin(x^2 −arctan((1/x^2 ))))/( (√(1 + x^4 )))) dx

$\underset{0}{\int^{\infty}} \frac{s i n (x^{2} - a r c t a n (\frac{1}{x^{2}}))}{\sqrt{1 + x^{4}}} d x$

Question Number 139288 Answers: 4 Comments: 0

IF ((x^3 +1)/(x^2 −1))=x+(√(6/x)) 6x^2 −5x+4=??

$I F \frac{x^{3} + 1}{x^{2} - 1} = x + \sqrt{\frac{6}{x}}$ $6 x^{2} - 5 x + 4 = ? ?$

Question Number 139286 Answers: 0 Comments: 1

(1/((1−x)^n ))=( ((n),(0) ))+( ((n),(1) ))x+( ((n),(2) ))x^2 +( ((n),(3) ))x^3 +...

$\frac{1}{{(1 - x)}^{n}} = ((\begin{matrix} n \\ 0 \end{matrix})) + ((\begin{matrix} n \\ 1 \end{matrix})) x + ((\begin{matrix} n \\ 2 \end{matrix})) x^{2} + ((\begin{matrix} n \\ 3 \end{matrix})) x^{3} + . . .$

Question Number 139285 Answers: 2 Comments: 0

∫_0 ^(π/2) sin^2 (x)cos^6 (x)dx

$\int_{0}^{\frac{π}{2}} {s i n}^{2} (x) {c o s}^{6} (x) d x$

Question Number 139283 Answers: 2 Comments: 0

∫_0 ^∞ ((ln^2 x)/(x^2 +a^2 ))dx=(π/(8a))(π^2 +4ln^2 a) ,a>0

$\int_{0}^{\infty} \frac{{l n}^{2} x}{x^{2} + a^{2}} d x = \frac{π}{8 a} (π^{2} + 4 {l n}^{2} a), a > 0$

Question Number 139282 Answers: 1 Comments: 0

∫_0 ^∞ ((ln x)/(x^2 +a^2 ))dx=(π/(2a))∙ln a ,a>0

$\int_{0}^{\infty} \frac{l n x}{x^{2} + a^{2}} d x = \frac{π}{2 a} \cdot l n a, a > 0$

Question Number 139276 Answers: 1 Comments: 1

Question Number 139278 Answers: 2 Comments: 0

(√((3000000 + 3000000)/(4000000 + 4000000))) = ?

$\sqrt{\frac{3000000 + 3000000}{4000000 + 4000000}} = ?$

Question Number 139273 Answers: 0 Comments: 0

Question Number 139248 Answers: 2 Comments: 0

∫_a ^( b) f(t)g′(t)dt= ?

$\int_{a}^{b} f (t) g^{'} (t) d t = ?$

Question Number 139232 Answers: 1 Comments: 4

A man has ′n′ pair of black shoes and ′m′ pairs of brown shoes. Man hurriedly wear two shoes. What is probability that both of them are black?

$A m a n h a s^{'} n^{'} p a i r o f b l a c k s h o e s a n d^{'} m^{'}$ $p a i r s o f b r o w n s h o e s . M a n h u r r i e d l y w e a r$ $t w o s h o e s . W h a t i s p r o b a b i l i t y t h a t b o t h o f$ $t h e m a r e b l a c k ?$

Question Number 139259 Answers: 1 Comments: 0

∫ ((cos x+(7)^(1/3) )/(sin x+(√6))) dx =?

$\int \frac{\cos x + \sqrt[3]{7}}{\sin x + \sqrt{6}} dx = ?$

Question Number 139257 Answers: 2 Comments: 0

Σ_(k=1) ^∞ (((−1)^k (2k−1))/k^2 ) =?

$\underset{k = 1}{\sum^{\infty}} \frac{{(- 1)}^{k} (2 k - 1)}{k^{2}} = ?$

Question Number 139255 Answers: 2 Comments: 0

How do you find a point on the curve y = x^2 that is closest to the point? (16, 1/2)

Question Number 139226 Answers: 0 Comments: 0

∫_0 ^1 e^x^2 dx=(e/(1+(2/(1+(4/(1+(6/(1+(8/(1+((10)/(1+...))))))))))))

$\int_{0}^{1} e^{x^{2}} d x = \frac{e}{1 + \frac{2}{1 + \frac{4}{1 + \frac{6}{1 + \frac{8}{1 + \frac{10}{1 + . . .}}}}}}$

Question Number 139224 Answers: 2 Comments: 2

𝛗=∫_0 ^( 1) ((ln(1+x+x^2 ))/x)dx φ=∫_0 ^( 1) ((ln(1−x^3 )−ln(1−x))/x)dx =Ω+(π^2 /6) Ω=−Σ∫_0 ^( 1) (x^(3n−1) /n)dx=−Σ_(n=1 ) ^∞ (1/(3n^2 )) =((−π^2 )/(18)) .....≽ 𝛗=(π^2 /9) .......(π^2 /9)−(π^2 /(24)) =((5π^2 )/(72))

$ϕ = \int_{0}^{1} \frac{l n (1 + x + x^{2})}{x} d x$ $ϕ = \int_{0}^{1} \frac{l n (1 - x^{3}) - l n (1 - x)}{x} d x$ $= Ω + \frac{π^{2}}{6}$ $Ω = - Σ \int_{0}^{1} \frac{x^{3 n - 1}}{n} d x = - \underset{n = 1}{\sum^{\infty}} \frac{1}{3 n^{2}}$ $= \frac{- π^{2}}{18} . . . . . ≽ ϕ = \frac{π^{2}}{9}$ $. . . . . . . \frac{π^{2}}{9} - \frac{π^{2}}{24} = \frac{5 π^{2}}{72}$

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