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Question Number 158517 Answers: 0 Comments: 0

ϑ =∫_( 0) ^( π/2) (dx/((1+(1/(sin^2 x)))^2 )) ?

$$\:\:\:\vartheta\:=\int_{\:\mathrm{0}} ^{\:\pi/\mathrm{2}} \frac{{dx}}{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} {x}}\right)^{\mathrm{2}} }\:? \\ $$

Question Number 158715 Answers: 0 Comments: 0

Question Number 158531 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (1/(n^2 (2n+1)^3 ))=?

$$\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} \left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)^{\mathrm{3}} }=? \\ $$

Question Number 158506 Answers: 0 Comments: 0

Question Number 158508 Answers: 0 Comments: 0

if x;y;z>0 and xyz=27 prove that: (1/(x^2 +27)) + (1/(y^2 +27)) + (1/(z^2 +27)) ≤ (1/(12))

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{xyz}=\mathrm{27}\:\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} +\mathrm{27}}\:+\:\frac{\mathrm{1}}{\mathrm{y}^{\mathrm{2}} +\mathrm{27}}\:+\:\frac{\mathrm{1}}{\mathrm{z}^{\mathrm{2}} +\mathrm{27}}\:\leqslant\:\frac{\mathrm{1}}{\mathrm{12}} \\ $$

Question Number 158503 Answers: 0 Comments: 5

Question Number 158590 Answers: 1 Comments: 0

prove that 1. I= ∫_0 ^( (π/2)) (( sin( x+tan(x)))/(sin(x)))dx =(π/2) 2. J = ∫_0 ^( (π/2)) ((sin(x−tan(x)))/(sin(x)))dx=((1/e) −(1/2))π

$$ \\ $$$$\:\:\:{prove}\:{that}\: \\ $$$$\mathrm{1}.\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{sin}\left(\:{x}+{tan}\left({x}\right)\right)}{{sin}\left({x}\right)}{dx}\:=\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{2}.\:\mathrm{J}\:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{sin}\left({x}−{tan}\left({x}\right)\right)}{{sin}\left({x}\right)}{dx}=\left(\frac{\mathrm{1}}{{e}}\:−\frac{\mathrm{1}}{\mathrm{2}}\right)\pi \\ $$$$ \\ $$

Question Number 158494 Answers: 0 Comments: 0

Question Number 158492 Answers: 1 Comments: 0

∫ (dx/(1+x^7 )) ?

$$\:\int\:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{7}} }\:? \\ $$

Question Number 158489 Answers: 0 Comments: 0

find the partial derivatives of the function with respect to each variable f(x,y)=∫_x ^y g(t) dt

$${find}\:{the}\:{partial}\:{derivatives}\:{of}\:{the}\:{function} \\ $$$${with}\:{respect}\:{to}\:{each}\:{variable} \\ $$$${f}\left({x},{y}\right)=\int_{{x}} ^{{y}} {g}\left({t}\right)\:{dt} \\ $$

Question Number 158483 Answers: 0 Comments: 0

Question Number 158482 Answers: 0 Comments: 0

Question Number 158519 Answers: 0 Comments: 0

Question Number 158477 Answers: 2 Comments: 0

I_(n ) =∫_0 ^1 (x^(2n+1) /( (√(1+x^2 ))))dx , n≥0 prove that ∀ n≥0 (2n+1)I_n =(√2)−2nI_(n−1)

$${I}_{{n}\:} =\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}{n}+\mathrm{1}} }{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx}\:,\:{n}\geqslant\mathrm{0}\: \\ $$$${prove}\:{that}\:\forall\:{n}\geqslant\mathrm{0}\: \\ $$$$\left(\mathrm{2}{n}+\mathrm{1}\right){I}_{{n}} =\sqrt{\mathrm{2}}−\mathrm{2}{nI}_{{n}−\mathrm{1}} \\ $$

Question Number 158529 Answers: 1 Comments: 0

In how many ways can 30 students be distributed to 10 schools, if 1. each school should get at least one student. 2. no restriction

$${In}\:{how}\:{many}\:{ways}\:{can}\:\mathrm{30}\:{students} \\ $$$${be}\:{distributed}\:{to}\:\mathrm{10}\:{schools},\:{if} \\ $$$$\mathrm{1}.\:{each}\:{school}\:{should}\:{get}\:{at}\:{least}\:{one} \\ $$$$\:\:\:\:\:{student}. \\ $$$$\mathrm{2}.\:{no}\:{restriction} \\ $$

Question Number 158528 Answers: 0 Comments: 0

Π_(n=1) ^∞ ((((n+1)^3 −1)/((n+1)^3 +1)))=?

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\prod}}\left(\frac{\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)^{\mathrm{3}} −\mathrm{1}}{\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)^{\mathrm{3}} +\mathrm{1}}\right)=? \\ $$

Question Number 158472 Answers: 1 Comments: 1

Question Number 158469 Answers: 2 Comments: 0

find the maclaurin series expension for the function f(x) = sin^2 x ; x_o = 0

$${find}\:{the}\:{maclaurin}\:{series}\:{expension} \\ $$$${for}\:{the}\:{function}\:{f}\left({x}\right)\:=\:{sin}^{\mathrm{2}} {x}\:;\:\:\:\:\:\:{x}_{{o}} =\:\mathrm{0} \\ $$

Question Number 158465 Answers: 1 Comments: 0

Question Number 158455 Answers: 1 Comments: 0

Question Number 158450 Answers: 0 Comments: 0

Demontrer que minN=0

$$\mathrm{Demontrer}\:\mathrm{que}\:\mathrm{min}\mathbb{N}=\mathrm{0} \\ $$

Question Number 158444 Answers: 1 Comments: 5

if x and y are positive integers with ((2010)/(2011)) < (x/y) < ((2011)/(2012)) then compute the minimum value for x+y and the values of x and y which achieves this minimum

$$\mathrm{if}\:\:\boldsymbol{\mathrm{x}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{y}}\:\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{with} \\ $$$$\frac{\mathrm{2010}}{\mathrm{2011}}\:<\:\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{y}}}\:<\:\frac{\mathrm{2011}}{\mathrm{2012}}\:\:\mathrm{then}\:\mathrm{compute}\:\mathrm{the} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:\mathrm{for}\:\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\:\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{values}\:\mathrm{of}\:\:\boldsymbol{\mathrm{x}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{y}}\:\:\mathrm{which}\:\mathrm{achieves} \\ $$$$\mathrm{this}\:\mathrm{minimum} \\ $$

Question Number 158443 Answers: 1 Comments: 0

How many divisors has the positive integer n which verify n^n = 2027^(2027^(2028) ) ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{divisors}\:\mathrm{has}\:\mathrm{the}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{which}\:\mathrm{verify} \\ $$$$\mathrm{n}^{\boldsymbol{\mathrm{n}}} \:=\:\mathrm{2027}^{\mathrm{2027}^{\mathrm{2028}} } \:? \\ $$

Question Number 158438 Answers: 1 Comments: 2

Question Number 158437 Answers: 0 Comments: 0

Question Number 158410 Answers: 2 Comments: 0

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