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Question Number 88071 Answers: 1 Comments: 7

Question Number 88069 Answers: 1 Comments: 2

Question Number 88068 Answers: 2 Comments: 3

Question Number 88067 Answers: 1 Comments: 0

Question Number 88065 Answers: 0 Comments: 3

lim_(n→∞) (1/n)Σ_(i=1) ^n cos^2 (((πi)/n))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{{n}}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\pi{i}}{{n}}\right) \\ $$

Question Number 88064 Answers: 1 Comments: 0

∫ (dx/(cos x(2+sin x)))?

$$\int\:\frac{\mathrm{dx}}{\mathrm{cos}\:\mathrm{x}\left(\mathrm{2}+\mathrm{sin}\:\mathrm{x}\right)}? \\ $$

Question Number 88050 Answers: 1 Comments: 6

Question Number 88045 Answers: 0 Comments: 2

∫ ((sin x)/(2sin x+cos x)) dx

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

Question Number 88042 Answers: 1 Comments: 0

find max and min value of function f(x) = (5/(−3cos x−4sin x))

$$\mathrm{find}\:\mathrm{max}\:\mathrm{and}\:\mathrm{min}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{5}}{−\mathrm{3cos}\:\mathrm{x}−\mathrm{4sin}\:\mathrm{x}} \\ $$

Question Number 88041 Answers: 1 Comments: 0

Question Number 88040 Answers: 0 Comments: 2

Find the max and min of function ((a+bsin x)/(b+asin x)) where b>a>0 in the interval 0≤x≤2π.sketch a=4 and b=5

$${Find}\:{the}\:{max}\:{and}\:{min} \\ $$$${of}\:{function} \\ $$$$\frac{{a}+{b}\mathrm{sin}\:{x}}{{b}+{a}\mathrm{sin}\:{x}} \\ $$$${where}\:{b}>{a}>\mathrm{0}\:{in}\:{the}\: \\ $$$${interval}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}\pi.{sketch} \\ $$$${a}=\mathrm{4}\:{and}\:{b}=\mathrm{5} \\ $$

Question Number 88039 Answers: 1 Comments: 0

Obtain the first four term of the expansion (4−x)^(1/3) when (1)∣x∣<1 (ii)∣x∣>1

$${Obtain}\:{the}\:{first}\:{four} \\ $$$${term}\:{of}\:{the}\:{expansion} \\ $$$$\left(\mathrm{4}−{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} {when} \\ $$$$\left(\mathrm{1}\right)\mid{x}\mid<\mathrm{1} \\ $$$$\left({ii}\right)\mid{x}\mid>\mathrm{1} \\ $$

Question Number 88033 Answers: 0 Comments: 4

find ∫_0 ^1 ((sin(x))/x)dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{sin}\left({x}\right)}{{x}}{dx} \\ $$

Question Number 88032 Answers: 0 Comments: 7

decompose inside R(x) the fraction 1) F(x) =(1/(x^3 (x−2)^3 )) 2) F(x) =(1/((x+1)^4 (x−3)^4 ))

$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$$\left.\mathrm{1}\right)\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{3}} \left({x}−\mathrm{2}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{4}} \left({x}−\mathrm{3}\right)^{\mathrm{4}} } \\ $$

Question Number 88029 Answers: 1 Comments: 0

Question Number 88027 Answers: 0 Comments: 1

Question Number 88026 Answers: 0 Comments: 0

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))

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

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