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Question Number 108384 Answers: 3 Comments: 0

(1)∫ ((√(sin x))/( (√(sin x)) + (√(cos x)))) dx ? (2) (d^2 y/dx^2 )−2(dy/dx) +y = e^x

$$\left(\mathrm{1}\right)\int\:\frac{\sqrt{\mathrm{sin}\:{x}}}{\:\sqrt{\mathrm{sin}\:{x}}\:+\:\sqrt{\mathrm{cos}\:{x}}}\:{dx}\:? \\ $$$$\left(\mathrm{2}\right)\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{2}\frac{{dy}}{{dx}}\:+{y}\:=\:{e}^{{x}} \\ $$

Question Number 108382 Answers: 2 Comments: 1

((bobhans)/(⋰⋱)) ∫_(π/2) ^π ∣ cos x−sin x ∣ dx ?

$$\:\:\:\frac{\boldsymbol{{bobhans}}}{\iddots\ddots} \\ $$$$\:\underset{\pi/\mathrm{2}} {\overset{\pi} {\int}}\mid\:\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\:\mid\:{dx}\:? \\ $$$$ \\ $$

Question Number 108377 Answers: 1 Comments: 0

There are 20 white and 10 black balls in the box.In turn draw 4 balls and each draw ball is returned to the box before the next ball is drawn and the balls in the box are mixed.What is the probability that out of four balls taken out there will be two white balls?

$$\mathrm{There}\:\mathrm{are}\:\mathrm{20}\:\mathrm{white}\:\mathrm{and}\:\mathrm{10}\:\mathrm{black}\:\mathrm{balls} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{box}.\mathrm{In}\:\mathrm{turn}\:\mathrm{draw}\:\mathrm{4}\:\mathrm{balls}\:\mathrm{and} \\ $$$$\mathrm{each}\:\mathrm{draw}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{returned}\:\mathrm{to}\:\mathrm{the}\:\mathrm{box} \\ $$$$\mathrm{before}\:\mathrm{the}\:\mathrm{next}\:\mathrm{ball}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\mathrm{balls}\:\mathrm{in}\:\mathrm{the}\:\mathrm{box}\:\mathrm{are}\:\mathrm{mixed}.\mathrm{What}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{probability}\:\mathrm{that}\:\mathrm{out}\:\mathrm{of}\:\mathrm{four}\:\mathrm{balls} \\ $$$$\mathrm{taken}\:\mathrm{out}\:\mathrm{there}\:\mathrm{will}\:\mathrm{be}\:\mathrm{two}\:\mathrm{white}\:\mathrm{balls}? \\ $$

Question Number 108453 Answers: 1 Comments: 0

If R=(7+4(√3))^(2n) = I+f, where I ∈ N and 0<f<1, then R(1−f) equals

$$\mathrm{If}\:\:{R}=\left(\mathrm{7}+\mathrm{4}\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}} \:=\:{I}+{f},\:\mathrm{where}\:{I}\:\in\:{N} \\ $$$$\mathrm{and}\:\:\:\mathrm{0}<{f}<\mathrm{1},\:\mathrm{then}\:{R}\left(\mathrm{1}−{f}\right)\:\mathrm{equals} \\ $$

Question Number 108370 Answers: 0 Comments: 1

Question Number 108369 Answers: 1 Comments: 0

Question Number 108366 Answers: 0 Comments: 1

Question Number 108365 Answers: 1 Comments: 0

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

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

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