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Question Number 64269 Answers: 1 Comments: 0

∫_(−π) ^π [cos px−sin qx]^2 dx, where p, q are integers is equal to

$$\underset{−\pi} {\overset{\pi} {\int}}\:\left[\mathrm{cos}\:{px}−\mathrm{sin}\:{qx}\right]^{\mathrm{2}} {dx},\:\mathrm{where}\:{p},\:{q}\:\mathrm{are}\: \\ $$$$\mathrm{integers}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 64268 Answers: 0 Comments: 1

lim_(n→∞) ((1^(99) +2^(99) +...+ n^(99) )/n^(100) ) =

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}^{\mathrm{99}} +\mathrm{2}^{\mathrm{99}} +...+\:{n}^{\mathrm{99}} }{{n}^{\mathrm{100}} }\:= \\ $$

Question Number 64270 Answers: 1 Comments: 0

∫((5sin(x) cos(x))/((cos(x)+1))^(1/3) ) dx

$$\int\frac{\mathrm{5}{sin}\left({x}\right)\:{cos}\left({x}\right)}{\sqrt[{\mathrm{3}}]{{cos}\left({x}\right)+\mathrm{1}}}\:{dx} \\ $$

Question Number 64204 Answers: 1 Comments: 1

Question Number 64203 Answers: 1 Comments: 5

M is the midpoint of segment AB. Prove that,for every point P in space, ∣PM∣ ≤ ((∣PA∣ +∣PB∣)/2)

$$\mathrm{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:\mathrm{segment}\:\mathrm{AB}.\:\mathrm{Prove} \\ $$$$\mathrm{that},\mathrm{for}\:\mathrm{every}\:\mathrm{point}\:\mathrm{P}\:\mathrm{in}\:\mathrm{space}, \\ $$$$\mid\mathrm{PM}\mid\:\leqslant\:\frac{\mid\mathrm{PA}\mid\:+\mid\mathrm{PB}\mid}{\mathrm{2}} \\ $$

Question Number 64202 Answers: 0 Comments: 0

Prove 385^(1980) +18^(1980) is not a square.

$$\mathrm{Prove}\:\mathrm{385}^{\mathrm{1980}} +\mathrm{18}^{\mathrm{1980}} \:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{square}.\: \\ $$

Question Number 64201 Answers: 2 Comments: 0

Find all positive solutions x+y+z = 1 x^3 +y^3 + z^3 + xyz = x^4 + y^4 + z^4 +1

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{solutions}\: \\ $$$$\mathrm{x}+\mathrm{y}+\mathrm{z}\:=\:\mathrm{1} \\ $$$$\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} +\:\mathrm{z}^{\mathrm{3}} \:+\:\mathrm{xyz}\:=\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{z}^{\mathrm{4}} \:+\mathrm{1} \\ $$

Question Number 64200 Answers: 1 Comments: 1

Question Number 64187 Answers: 1 Comments: 0

Question Number 64186 Answers: 1 Comments: 0

Question Number 64185 Answers: 2 Comments: 3

Question Number 64176 Answers: 1 Comments: 0

Question Number 64175 Answers: 0 Comments: 1

m^3 −4m

$${m}^{\mathrm{3}} \:−\mathrm{4}{m}\: \\ $$

Question Number 64174 Answers: 0 Comments: 0

2^n / 5^2^n + 1 infinite series sum ffom 0 to infinity

$$\mathrm{2}^{{n}} \:/\:\mathrm{5}^{\mathrm{2}^{{n}} } \:+\:\mathrm{1}\:{infinite}\:{series}\:{sum}\:{ffom}\:\mathrm{0}\:{to}\: \\ $$$${infinity} \\ $$

Question Number 64173 Answers: 0 Comments: 1

show tbat Π_(k=1) ^n (1−((sin^2 ((θ/(2n))))/(sin^2 (((2k−1)/4)π))))=cos(θ)

$${show}\:{tbat}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}−\frac{{sin}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}{n}}\right)}{{sin}^{\mathrm{2}} \left(\frac{\mathrm{2}{k}−\mathrm{1}}{\mathrm{4}}\pi\right)}\right)={cos}\left(\theta\right) \\ $$

Question Number 64166 Answers: 1 Comments: 0

calculate A_n =∫_(−∞) ^(+∞) (dx/((x^2 +1)(x^2 +2)....(x^2 +n))) with n integr natural and n≥1

$${calculate}\:\:{A}_{{n}} =\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} \:+\mathrm{2}\right)....\left({x}^{\mathrm{2}} \:+{n}\right)} \\ $$$${with}\:{n}\:{integr}\:{natural}\:\:{and}\:{n}\geqslant\mathrm{1} \\ $$

Question Number 64160 Answers: 2 Comments: 5

let f(x) =∫_0 ^1 (dt/(x +2^t )) with x>0 1) determine a explicit form for f(x) 2) determine also g(x)=∫_0 ^1 (dt/((x+2^x )^2 )) 3) give f^((n)) (x) at form of integral 4) calculate ∫_0 ^1 (dt/(1+2^t )) and ∫_0 ^1 (dt/((1+2^t )^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{{x}\:+\mathrm{2}^{{t}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left({x}+\mathrm{2}^{{x}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}\: \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\mathrm{1}+\mathrm{2}^{{t}} }\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left(\mathrm{1}+\mathrm{2}^{{t}} \right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 64159 Answers: 1 Comments: 1

calculate ∫_0 ^1 (dx/(3+2^x ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{3}+\mathrm{2}^{{x}} } \\ $$

Question Number 64158 Answers: 0 Comments: 0

Question Number 64153 Answers: 1 Comments: 0

solve to z^2 x^2 −y^2 =24

$${solve}\:{to}\:{z}^{\mathrm{2}} \:\:\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{24} \\ $$

Question Number 64150 Answers: 1 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +1)(x^2 +2)(x^2 +3)))

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} +\mathrm{2}\right)\left({x}^{\mathrm{2}} +\mathrm{3}\right)} \\ $$

Question Number 64147 Answers: 2 Comments: 0

Question Number 64139 Answers: 1 Comments: 2

∫_( 0) ^1 (dx/(x^2 +2x cos α+1)) = α sin α

$$\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{2}{x}\:\mathrm{cos}\:\alpha+\mathrm{1}}\:=\:\alpha\:\mathrm{sin}\:\alpha \\ $$

Question Number 64138 Answers: 1 Comments: 0

Question Number 64130 Answers: 2 Comments: 1

(√(4x+((12)/x)))=((x^2 +7)/(x+1)) x=?

$$\sqrt{\mathrm{4}\boldsymbol{\mathrm{x}}+\frac{\mathrm{12}}{\boldsymbol{\mathrm{x}}}}=\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{7}}{\boldsymbol{\mathrm{x}}+\mathrm{1}} \\ $$$$\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 64129 Answers: 0 Comments: 0

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