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Question Number 87534 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) ((arctan(sinx))/(sinx))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{arctan}\left({sinx}\right)}{{sinx}}{dx} \\ $$

Question Number 87533 Answers: 3 Comments: 2

Question Number 87532 Answers: 0 Comments: 2

(1).Find the general solution: y= px +p^n (2).Solve the differential equation: (x+1)^2 (d^2 y/dx^2 ) + (x+1)(dy/dx)= (2x+3)(2x+4).

Question Number 87530 Answers: 0 Comments: 1

1) calculate U_n =∫_0 ^∞ e^(−n[x]) sin(((πx)/n))dx nnatural and n≥1 2)determine nature of Σ U_n

$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{e}^{−{n}\left[{x}\right]} {sin}\left(\frac{\pi{x}}{{n}}\right){dx}\:\:{nnatural}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right){determine}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 87527 Answers: 0 Comments: 1

find ∫_0 ^∞ ((arctan(3x))/(x^2 +x+1))dx

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{3}{x}\right)}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx} \\ $$

Question Number 87526 Answers: 0 Comments: 1

calculate ∫_0 ^∞ e^(−[nx]) dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left[{nx}\right]} \:{dx} \\ $$

Question Number 87516 Answers: 1 Comments: 0

{ ((2^((x+y)/3) +2^((x+y)/6) =6)),((x^2 +5y^2 =6xy)) :}

$$\begin{cases}{\mathrm{2}^{\frac{{x}+{y}}{\mathrm{3}}} +\mathrm{2}^{\frac{{x}+{y}}{\mathrm{6}}} =\mathrm{6}}\\{{x}^{\mathrm{2}} +\mathrm{5}{y}^{\mathrm{2}} =\mathrm{6}{xy}}\end{cases} \\ $$

Question Number 87511 Answers: 2 Comments: 0

Question Number 87508 Answers: 0 Comments: 1

Question Number 87505 Answers: 1 Comments: 6

Question Number 87504 Answers: 1 Comments: 0

solve (dy/dx)=2(((2+y)/(1+x+y)))^2

$${solve}\: \\ $$$$\frac{{dy}}{{dx}}=\mathrm{2}\left(\frac{\mathrm{2}+{y}}{\mathrm{1}+{x}+{y}}\right)^{\mathrm{2}} \\ $$

Question Number 87503 Answers: 1 Comments: 4

∫(x^2 /(1+x^5 ))dx

$$\int\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{5}} }{dx} \\ $$

Question Number 87497 Answers: 0 Comments: 0

A complex number z is defined by z = (1/2)(cos θ + isin θ),such that z^n = (1/2^n ) (cos nθ + isin nθ) Using De Moivre′s theorem,or otherwise, show that (i) Σ_(r=0) ^∞ (1/4^r ) sin 2rθ is a convergent geometic progression. (ii) Σ_(r=0) ^∞ (1/4^r ) sin 2r = ((14 sin 2θ)/(17−16cos 2θ))

$$\mathrm{A}\:\mathrm{complex}\:\mathrm{number}\:{z}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{z}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\theta\:+\:{i}\mathrm{sin}\:\theta\right),\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}^{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\:\left(\mathrm{cos}\:{n}\theta\:+\:{i}\mathrm{sin}\:{n}\theta\right) \\ $$$$\mathrm{Using}\:\mathrm{De}\:\mathrm{Moivre}'\mathrm{s}\:\mathrm{theorem},\mathrm{or}\:\mathrm{otherwise},\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\left(\mathrm{i}\right)\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{4}^{{r}} }\:\mathrm{sin}\:\mathrm{2r}\theta\:\mathrm{is}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{geometic}\:\mathrm{progression}. \\ $$$$\left(\mathrm{ii}\right)\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{4}^{{r}} }\:\mathrm{sin}\:\mathrm{2}{r}\:=\:\frac{\mathrm{14}\:\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{17}−\mathrm{16cos}\:\mathrm{2}\theta} \\ $$

Question Number 87854 Answers: 0 Comments: 3

∫ (1/(sin x+2cos x+3)) dx

$$\int\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}+\mathrm{2cos}\:\mathrm{x}+\mathrm{3}}\:\mathrm{dx} \\ $$

Question Number 87492 Answers: 1 Comments: 0

((2+3^2 )/(1!+2!+3!+4!))+((3+4^2 )/(2!+3!+4!+5!))+...+((2013+2014^2 )/(2012!+2013!+2014!+2015!))

$$\frac{\mathrm{2}+\mathrm{3}^{\mathrm{2}} }{\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+\mathrm{4}!}+\frac{\mathrm{3}+\mathrm{4}^{\mathrm{2}} }{\mathrm{2}!+\mathrm{3}!+\mathrm{4}!+\mathrm{5}!}+...+\frac{\mathrm{2013}+\mathrm{2014}^{\mathrm{2}} }{\mathrm{2012}!+\mathrm{2013}!+\mathrm{2014}!+\mathrm{2015}!} \\ $$

Question Number 87488 Answers: 2 Comments: 1

sin^4 x + sin^4 (x+(π/4)) = (1/4) x ∈ [ 0,2π ]

$$\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\:+\:\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}+\frac{\pi}{\mathrm{4}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\mathrm{x}\:\in\:\left[\:\mathrm{0},\mathrm{2}\pi\:\right]\: \\ $$

Question Number 90465 Answers: 0 Comments: 6