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Question Number 88021 Answers: 1 Comments: 2

Question Number 88015 Answers: 0 Comments: 0

if u=f(x,y) where x=rcos(θ) , y=r sin(θ) prove ((∂u/∂x))^2 +((∂u/∂y))^2 =((∂u/∂r))^2 +(1/r)((∂u/∂θ))^2

$${if}\:{u}={f}\left({x},{y}\right)\:{where}\:{x}={rcos}\left(\theta\right)\:\:,\:{y}={r}\:{sin}\left(\theta\right) \\ $$$${prove}\: \\ $$$$\left(\frac{\partial{u}}{\partial{x}}\right)^{\mathrm{2}} +\left(\frac{\partial{u}}{\partial{y}}\right)^{\mathrm{2}} =\left(\frac{\partial{u}}{\partial{r}}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{{r}}\left(\frac{\partial{u}}{\partial\theta}\right)^{\mathrm{2}} \\ $$

Question Number 88014 Answers: 2 Comments: 0

If there is no second′s hand on a clock and the minute and hour hand move in continuous fashion, then exactly at what time between 02:10 and 02:15 does the position of the two hands exactly coincide?

$${If}\:{there}\:{is}\:{no}\:{second}'{s}\:{hand}\:{on} \\ $$$${a}\:{clock}\:{and}\:{the}\:{minute}\:{and}\:{hour} \\ $$$${hand}\:{move}\:{in}\:{continuous}\:{fashion}, \\ $$$${then}\:{exactly}\:{at}\:{what}\:{time}\:{between} \\ $$$$\mathrm{02}:\mathrm{10}\:\:{and}\:\mathrm{02}:\mathrm{15}\:{does}\:{the}\:{position} \\ $$$${of}\:{the}\:{two}\:{hands}\:{exactly}\:{coincide}? \\ $$

Question Number 88010 Answers: 0 Comments: 2

∫((x^2 +2x+3)/(√(x^2 +x+1)))dx

$$\int\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}}{\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$

Question Number 88007 Answers: 1 Comments: 0

∫_0 ^1 (√((√((4/x)−3))−1))dx=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\sqrt{\sqrt{\frac{\mathrm{4}}{{x}}−\mathrm{3}}−\mathrm{1}}{dx}=? \\ $$

Question Number 88004 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (H_n /(2n+1))(π^2 +2H_n ^((2)) −8H_(2n) ^((2)) ) =((83)/4)ζ(4)−7log(2)ζ(3)−8log^2 (2)ζ(2)−(2/3)log^4 (2)−16 Li_4 ((1/2)) where H_n ^((m)) =1+(1/2^m )+.....+(1/n^m ) represents the nth generalized harmonic number of order m , ζ denotes the Riemann zeta function,and Li_n designates the poly logarithm

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{H}_{{n}} }{\mathrm{2}{n}+\mathrm{1}}\left(\pi^{\mathrm{2}} +\mathrm{2}{H}_{{n}} ^{\left(\mathrm{2}\right)} −\mathrm{8}{H}_{\mathrm{2}{n}} ^{\left(\mathrm{2}\right)} \right) \\ $$$$=\frac{\mathrm{83}}{\mathrm{4}}\zeta\left(\mathrm{4}\right)−\mathrm{7}{log}\left(\mathrm{2}\right)\zeta\left(\mathrm{3}\right)−\mathrm{8}{log}^{\mathrm{2}} \left(\mathrm{2}\right)\zeta\left(\mathrm{2}\right)−\frac{\mathrm{2}}{\mathrm{3}}{log}^{\mathrm{4}} \left(\mathrm{2}\right)−\mathrm{16}\:{Li}_{\mathrm{4}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$${where}\:{H}_{{n}} ^{\left({m}\right)} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{{m}} }+.....+\frac{\mathrm{1}}{{n}^{{m}} }\:{represents}\:{the}\:{nth}\:{generalized} \\ $$$${harmonic}\:{number}\:{of}\:{order}\:{m}\:,\:\zeta\:{denotes}\:{the}\:{Riemann} \\ $$$${zeta}\:{function},{and}\:{Li}_{{n}} {designates}\:{the}\:{poly}\:{logarithm} \\ $$

Question Number 88003 Answers: 0 Comments: 2

Determine all functions f[0,1]→Ω such that ∀x∈[0,1] f ′(x)+f(x)=f(0)+f(1)

$${Determine}\:{all}\:{functions}\:{f}\left[\mathrm{0},\mathrm{1}\right]\rightarrow\Omega \\ $$$${such}\:{that}\:\forall{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{f}\:'\left({x}\right)+{f}\left({x}\right)={f}\left(\mathrm{0}\right)+{f}\left(\mathrm{1}\right) \\ $$

Question Number 88000 Answers: 0 Comments: 2

Is there a formula to calculate Σ_(i=1) ^n (1/i^2 ) interms of n..?

$${Is}\:{there}\:{a}\:{formula}\:{to}\:{calculate}\: \\ $$$$ \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{i}^{\mathrm{2}} } \\ $$$${interms}\:{of}\:{n}..? \\ $$

Question Number 87998 Answers: 0 Comments: 0

calculate Σ_(n=0) ^∞ (((−1)^n )/(n^3 +1))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} \:+\mathrm{1}} \\ $$

Question Number 87996 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(3x)−arctanx)/x)dx