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Question Number 108350 Answers: 3 Comments: 0

Calculate (d^n /dx^n )(sinx) , (d^n /dx^n )(lnx)

$$\mathrm{Calculate}\:\frac{\mathrm{d}^{\mathrm{n}} }{\mathrm{dx}^{\mathrm{n}} }\left(\mathrm{sinx}\right)\:,\:\frac{\mathrm{d}^{\mathrm{n}} }{\mathrm{dx}^{\mathrm{n}} }\left(\mathrm{lnx}\right) \\ $$

Question Number 108349 Answers: 0 Comments: 0

Derive Leibniz′s formula : (fg)^((n)) (x_0 )=Σ_(k=0) ^n C_n ^k f^((k)) (x_0 )g^((n−k)) (x_0 )

$$\mathrm{Derive}\:\mathrm{Leibniz}'\mathrm{s}\:\mathrm{formula}\:: \\ $$$$\left(\mathrm{fg}\right)^{\left(\mathrm{n}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \mathrm{f}^{\left(\mathrm{k}\right)} \left(\mathrm{x}_{\mathrm{0}} \right)\mathrm{g}^{\left(\mathrm{n}−\mathrm{k}\right)} \left(\mathrm{x}_{\mathrm{0}} \right) \\ $$

Question Number 108371 Answers: 2 Comments: 0

prove by reccurence (1/2)+(1/4)+(1/6)+...+(1/(2n))<or =(n/2) thanks

$${prove}\:{by}\:{reccurence}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{6}}+...+\frac{\mathrm{1}}{\mathrm{2}{n}}<{or}\:=\frac{{n}}{\mathrm{2}} \\ $$$${thanks} \\ $$

Question Number 108341 Answers: 0 Comments: 0

lim_(x→+∞) {ln(cosh x) − x}

$$\underset{\mathrm{x}\rightarrow+\infty} {\mathrm{lim}}\left\{\mathrm{ln}\left(\mathrm{cosh}\:\mathrm{x}\right)\:−\:\mathrm{x}\right\} \\ $$

Question Number 108334 Answers: 1 Comments: 0

((⊸JS⊸)/△) If a random variable X follows normal distribution with mean 30 and variance 25. What is the probalility of X is less than 28 ?

$$\:\:\:\frac{\multimap{JS}\multimap}{\bigtriangleup} \\ $$$${If}\:{a}\:{random}\:{variable}\:{X}\:{follows}\:{normal} \\ $$$${distribution}\:{with}\:{mean}\:\mathrm{30}\:{and}\: \\ $$$${variance}\:\mathrm{25}.\:{What}\:{is}\:{the}\:{probalility} \\ $$$${of}\:{X}\:{is}\:{less}\:{than}\:\mathrm{28}\:? \\ $$

Question Number 108332 Answers: 1 Comments: 0

Question Number 108331 Answers: 1 Comments: 0

Question Number 108326 Answers: 1 Comments: 1

Question Number 108320 Answers: 1 Comments: 0

Question Number 108325 Answers: 0 Comments: 1

((⊸JS⊸)/(−−−−)) ∫ (dx/(x^8 (x^2 +1))) = ?

$$\:\:\:\frac{\multimap{JS}\multimap}{−−−−} \\ $$$$\int\:\frac{{dx}}{{x}^{\mathrm{8}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:=\:? \\ $$

Question Number 109063 Answers: 0 Comments: 1

Question Number 108319 Answers: 1 Comments: 0

find general solution x (dy/dx) + 3xy = 6

$$\:{find}\:{general}\:{solution}\: \\ $$$$\:\:\:\:\:\:{x}\:\frac{{dy}}{{dx}}\:+\:\mathrm{3}{xy}\:=\:\mathrm{6} \\ $$

Question Number 108309 Answers: 2 Comments: 0

((△BeMath△)/∴) Given (√(5+(√(9+2(√(15)))))) +(√(5−(√(9+2(√(15)))))) = x find the value of (x−(1/x))^2

$$\:\:\:\:\:\frac{\bigtriangleup\mathcal{B}{e}\mathcal{M}{ath}\bigtriangleup}{\therefore} \\ $$$${Given}\:\sqrt{\mathrm{5}+\sqrt{\mathrm{9}+\mathrm{2}\sqrt{\mathrm{15}}}}\:+\sqrt{\mathrm{5}−\sqrt{\mathrm{9}+\mathrm{2}\sqrt{\mathrm{15}}}}\:=\:{x} \\ $$$${find}\:{the}\:{value}\:{of}\:\left({x}−\frac{\mathrm{1}}{{x}}\right)^{\mathrm{2}} \\ $$

Question Number 108306 Answers: 0 Comments: 0

proof that ∫_(0) ^(π/4) ((sin^2 xcos^2 x)/((sin^3 x+cos^3 x)^2 )) dx =(1/6)

$$\mathrm{proof}\:\mathrm{that}\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\frac{{sin}^{\mathrm{2}} {xcos}^{\mathrm{2}} {x}}{\left({sin}^{\mathrm{3}} {x}+{cos}^{\mathrm{3}} {x}\right)^{\mathrm{2}} }\:{dx}\:=\frac{\mathrm{1}}{\mathrm{6}} \\ $$

Question Number 108305 Answers: 0 Comments: 0

proof that∫_(−∞) ^∞ (dx/((x^2 +ax+a^2 )(x^2 +bx+b^2 ))) equals ((2π(a+b))/((√)3ab(a^2 +ab+b^2 ))).

$${proof}\:{that}\underset{−\infty} {\overset{\infty} {\int}}\frac{{dx}}{\left({x}^{\mathrm{2}} +{ax}+{a}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} +{bx}+{b}^{\mathrm{2}} \right)}\:{equals}\:\:\frac{\mathrm{2}\pi\left(\mathrm{a}+\mathrm{b}\right)}{\sqrt{}\mathrm{3ab}\left(\mathrm{a}^{\mathrm{2}} +\mathrm{ab}+\mathrm{b}^{\mathrm{2}} \right)}. \\ $$

Question Number 108303 Answers: 1 Comments: 0

∫_0 ^(π/4) (1/((sinx+cosx)))sin^6 x dx equals

$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\frac{\mathrm{1}}{\left({sinx}+{cosx}\right)}{sin}^{\mathrm{6}} {x}\:{dx}\:{equals} \\ $$

Question Number 108298 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (((−1)^n )/(n^3 +1))= ?

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

Question Number 108295 Answers: 3 Comments: 0

((BobHans)/(βo♭)) (1) { (((x+y)(x^2 −y^2 ) = 9)),(((x−y)(x^2 +y^2 ) = 5)) :} find the solution (2) x (dy/dx) = x^2 +y^2 when x=1 give y = 2

$$\:\:\:\:\frac{\mathbb{B}\mathrm{ob}\mathbb{H}\mathrm{ans}}{\beta\mathrm{o}\flat} \\ $$$$\:\left(\mathrm{1}\right)\begin{cases}{\left(\mathrm{x}+\mathrm{y}\right)\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right)\:=\:\mathrm{9}}\\{\left(\mathrm{x}−\mathrm{y}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)\:=\:\mathrm{5}}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{x}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:\mathrm{when}\:\mathrm{x}=\mathrm{1}\:\mathrm{give}\:\mathrm{y}\:=\:\mathrm{2}\: \\ $$

Question Number 108291 Answers: 2 Comments: 0

((∥ BeMath ∥)/(°∫ dx°)) (1) Given (x+(√(1+x^2 )))(y+(√(1+y^2 )))=1 find (x+y)^2

$$\:\:\:\frac{\parallel\:\mathcal{B}{e}\mathcal{M}{ath}\:\parallel}{°\int\:{dx}°} \\ $$$$\left(\mathrm{1}\right)\:{Given}\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)\left({y}+\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }\right)=\mathrm{1} \\ $$$${find}\:\left({x}+{y}\right)^{\mathrm{2}} \: \\ $$

Question Number 108290 Answers: 0 Comments: 3

Why do all active content of forum of dates 12,13,14was disappear on my phone?

$$\mathrm{Why}\:\mathrm{do}\:\mathrm{all}\:\mathrm{active}\:\mathrm{content}\:\mathrm{of}\:\mathrm{forum}\:\mathrm{of} \\ $$$$\mathrm{dates}\:\mathrm{12},\mathrm{13},\mathrm{14was}\:\mathrm{disappear}\:\mathrm{on}\:\mathrm{my} \\ $$$$\mathrm{phone}? \\ $$

Question Number 108316 Answers: 3 Comments: 0

((▽BeMath▽)/△) If x^4 +x^2 = ((11)/5) , find the value of Ω = (((x+1)/(x−1)))^(1/3) + (((x−1)/(x+1)))^(1/3)

$$\:\:\:\:\frac{\bigtriangledown\mathcal{B}{e}\mathcal{M}{ath}\bigtriangledown}{\bigtriangleup} \\ $$$${If}\:{x}^{\mathrm{4}} +{x}^{\mathrm{2}} \:=\:\frac{\mathrm{11}}{\mathrm{5}}\:,\:{find}\:{the}\:{value}\:{of} \\ $$$$\Omega\:=\:\sqrt[{\mathrm{3}}]{\frac{{x}+\mathrm{1}}{{x}−\mathrm{1}}}\:+\:\sqrt[{\mathrm{3}}]{\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}}\: \\ $$

Question Number 108285 Answers: 0 Comments: 0

What is the nature of the integral ∫_1 ^∞ ((t^5 +3t+1)/(t^3 +100))e^(−t) dt

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{t}^{\mathrm{5}} +\mathrm{3t}+\mathrm{1}}{\mathrm{t}^{\mathrm{3}} +\mathrm{100}}\mathrm{e}^{−\mathrm{t}} \mathrm{dt} \\ $$

Question Number 108283 Answers: 1 Comments: 0

Given the function Γ defined by Γ(x)=∫_0 ^(+∞) t^(x−1) e^(−t) dt 1. What is the domain of definition of Γ ? 2. Show that ∀x∈ DΓ, xΓ(x)=Γ(x+1) and deduce the value of Γ(n), n∈N^∗ 3. Assuming ∫_0 ^(+∞) e^(−u^2 ) =((√π)/2), calculate Γ((1/2)) and deduce that Γ(n+(1/2))=(((2n)!(√π))/(2^2^n n!))

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{function}\:\Gamma\:\mathrm{defined}\:\mathrm{by}\:\Gamma\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{+\infty} \mathrm{t}^{\mathrm{x}−\mathrm{1}} \mathrm{e}^{−\mathrm{t}} \mathrm{dt} \\ $$$$\mathrm{1}.\:\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{definition}\:\mathrm{of}\:\Gamma\:? \\ $$$$\mathrm{2}.\:\:\mathrm{Show}\:\mathrm{that}\:\forall\mathrm{x}\in\:\mathrm{D}\Gamma,\:\mathrm{x}\Gamma\left(\mathrm{x}\right)=\Gamma\left(\mathrm{x}+\mathrm{1}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\Gamma\left(\mathrm{n}\right),\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{3}.\:\:\mathrm{Assuming}\:\int_{\mathrm{0}} ^{+\infty} \mathrm{e}^{−\mathrm{u}^{\mathrm{2}} } =\frac{\sqrt{\pi}}{\mathrm{2}},\:\mathrm{calculate}\:\Gamma\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that} \\ $$$$\:\:\:\:\:\Gamma\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\left(\mathrm{2n}\right)!\sqrt{\pi}}{\mathrm{2}^{\mathrm{2}^{\mathrm{n}} } \mathrm{n}!} \\ $$

Question Number 108282 Answers: 0 Comments: 0

Determine the nature of the integral ∫_2 ^(+∞) (√(t^2 +3t)) ln(cos((1/t))) sin^2 ((1/(ln(t))))dt

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int_{\mathrm{2}} ^{+\infty} \sqrt{\mathrm{t}^{\mathrm{2}} +\mathrm{3t}}\:\mathrm{ln}\left(\mathrm{cos}\left(\frac{\mathrm{1}}{\mathrm{t}}\right)\right)\:\mathrm{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{ln}\left(\mathrm{t}\right)}\right)\mathrm{dt} \\ $$

Question Number 108270 Answers: 0 Comments: 1

If x, y, z > −1, show that ((1 + x^2 )/(1 + y + z^2 )) + ((1 + y^2 )/(1 + z + x^2 )) + ((1 + z^2 )/(1 + x + y^2 )) ≥ 2

$$\mathrm{If}\:{x},\:{y},\:{z}\:>\:−\mathrm{1},\:\mathrm{show}\:\mathrm{that}\:\frac{\mathrm{1}\:+\:{x}^{\mathrm{2}} }{\mathrm{1}\:+\:{y}\:+\:{z}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}\:+\:{y}^{\mathrm{2}} }{\mathrm{1}\:+\:{z}\:+\:{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}\:+\:{z}^{\mathrm{2}} }{\mathrm{1}\:+\:{x}\:+\:{y}^{\mathrm{2}} }\:\geqslant\:\mathrm{2} \\ $$

Question Number 108265 Answers: 1 Comments: 1

lim_(x→0) ((10^x −1)/x^(10) ) defin or not defin???

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{\mathrm{10}^{{x}} −\mathrm{1}}{{x}^{\mathrm{10}} }\:\:\:\:\:{defin}\:{or}\:{not}\:{defin}??? \\ $$

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