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Question Number 35036    Answers: 1   Comments: 0

cslculate Σ_(k=1) ^n k^2 (n+1−k)

$${cslculate}\:\sum_{{k}=\mathrm{1}} ^{{n}} {k}^{\mathrm{2}} \left({n}+\mathrm{1}−{k}\right) \\ $$

Question Number 35035    Answers: 1   Comments: 0

calculate u_n = Σ_(k=1) ^n (k/((k+1)!))

$${calculate}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}}{\left({k}+\mathrm{1}\right)!} \\ $$

Question Number 35032    Answers: 0   Comments: 0

Question Number 34892    Answers: 0   Comments: 0

Question Number 34891    Answers: 1   Comments: 0

Question Number 34888    Answers: 1   Comments: 1

Question Number 34901    Answers: 0   Comments: 3

∫_(−π/2) ^(+π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =[−((2cos^((2n+1)/2) x)/(2n+1))]_(−π/2) ^(+π/2) =0? What is the mistake in above? ∫_(−π/2) ^(+π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =2∫_0 ^(π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =(4/(2n+1)) (this is correct answer)

$$\int_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\left[−\frac{\mathrm{2cos}^{\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}}} {x}}{\mathrm{2}{n}+\mathrm{1}}\right]_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} =\mathrm{0}? \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mistake}\:\mathrm{in}\:\mathrm{above}? \\ $$$$\int_{−\pi/\mathrm{2}} ^{+\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \sqrt{\mathrm{cos}^{\mathrm{2}{n}−\mathrm{1}} {x}−\mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {x}}{dx} \\ $$$$=\frac{\mathrm{4}}{\mathrm{2}{n}+\mathrm{1}}\:\left(\mathrm{this}\:\mathrm{is}\:\mathrm{correct}\:\mathrm{answer}\right) \\ $$

Question Number 34878    Answers: 1   Comments: 0

show that C_(n−1) ^(n+r−1) =C_r ^(n+r−1)

$${show}\:{that} \\ $$$$\overset{{n}+{r}−\mathrm{1}} {{C}}_{{n}−\mathrm{1}} =\overset{{n}+{r}−\mathrm{1}} {{C}}_{{r}} \\ $$

Question Number 34877    Answers: 2   Comments: 1

4 couples are to take a photograph with a newly wedded couple in a wedding party.In how many ways can this be done if: i)the celebrated couple must stand in the middle ii)each couple must stand next to each other iii)the celebrated couple must not stand next to each other

$$\mathrm{4}\:{couples}\:{are}\:{to}\:{take}\:{a}\:{photograph} \\ $$$${with}\:{a}\:{newly}\:{wedded}\:{couple}\:{in}\:{a} \\ $$$${wedding}\:{party}.{In}\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}\:{if}: \\ $$$$\left.{i}\right){the}\:{celebrated}\:{couple}\:{must}\:{stand} \\ $$$${in}\:{the}\:{middle} \\ $$$$\left.{ii}\right){each}\:{couple}\:{must}\:{stand}\:{next}\:{to} \\ $$$${each}\:{other} \\ $$$$\left.{iii}\right){the}\:{celebrated}\:{couple}\:{must}\:{not} \\ $$$${stand}\:{next}\:{to}\:{each}\:{other} \\ $$

Question Number 34870    Answers: 3   Comments: 4

Question Number 34866    Answers: 0   Comments: 0

find f(x)=∫_0 ^∞ ((arctan(x(t +(1/t))))/(1+t^2 ))dt

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({x}\left({t}\:+\frac{\mathrm{1}}{{t}}\right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\: \\ $$

Question Number 34865    Answers: 0   Comments: 0

let f(x)= e^(−(√(1+2x))) developp f at integr serie .

$${let}\:{f}\left({x}\right)=\:{e}^{−\sqrt{\mathrm{1}+\mathrm{2}{x}}} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 34864    Answers: 0   Comments: 0

let f(x)=x arctan(1+e^(−x) ) developp f at intrgr serie .

$${let}\:{f}\left({x}\right)={x}\:{arctan}\left(\mathrm{1}+{e}^{−{x}} \right) \\ $$$${developp}\:{f}\:{at}\:{intrgr}\:{serie}\:. \\ $$

Question Number 34863    Answers: 0   Comments: 1

let f(x)= ((artan(x+1))/(1+2x)) developp f at integr serie .

$${let}\:{f}\left({x}\right)=\:\frac{{artan}\left({x}+\mathrm{1}\right)}{\mathrm{1}+\mathrm{2}{x}} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 34862    Answers: 2   Comments: 8

find the value of f(x) = ∫_0 ^π ((cosx)/(1+2sin(2x)))dx

$${find}\:{the}\:{value}\:{of} \\ $$$${f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{cosx}}{\mathrm{1}+\mathrm{2}{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 34850    Answers: 2   Comments: 0

Question Number 34849    Answers: 0   Comments: 2

let f(x) = (e^(−x) /(2+x)) developp f at integr serie.

$${let}\:{f}\left({x}\right)\:=\:\:\:\:\frac{{e}^{−{x}} }{\mathrm{2}+{x}} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 34843    Answers: 0   Comments: 1

lim_ _(x→∞) ((ln x)/x) = ? You can only use series expansion / sandwich theorem!

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}_{} }\:\frac{\mathrm{ln}\:{x}}{{x}}\:=\:? \\ $$$${You}\:{can}\:\:{only}\:{use} \\ $$$${series}\:{expansion}\:/\:{sandwich}\:{theorem}! \\ $$

Question Number 34827    Answers: 1   Comments: 5

Find ∫ Sin^6 x dx

$$\boldsymbol{{Find}}\:\int\:\boldsymbol{{Sin}}^{\mathrm{6}} \boldsymbol{{x}}\:\boldsymbol{{dx}} \\ $$$$ \\ $$

Question Number 34821    Answers: 2   Comments: 1

Find range of y=(x/((x−1)(x−2))) .

$${Find}\:{range}\:{of} \\ $$$$\:\:\:{y}=\frac{{x}}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)}\:. \\ $$

Question Number 34792    Answers: 2   Comments: 0

solve for x 4x=2^x

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{x}} \\ $$$$\mathrm{4}\boldsymbol{{x}}=\mathrm{2}^{\boldsymbol{{x}}} \\ $$

Question Number 34845    Answers: 0   Comments: 0

f(x)=cos(x) g(x)=2^x f:R→R g:R→R^+ [f(x)]^2 +[f(π/2−x)]^2 =((log_2 g(x))/x);x≠0 f(g(x)x)=[f(g(x−1)x)]^2 +[f(π/2−g(x−1)x)]^2 find f and g

$${f}\left({x}\right)=\mathrm{cos}\left({x}\right) \\ $$$${g}\left({x}\right)=\mathrm{2}^{{x}} \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${g}:\mathbb{R}\rightarrow\mathbb{R}^{+} \\ $$$$\left[{f}\left({x}\right)\right]^{\mathrm{2}} +\left[{f}\left(\pi/\mathrm{2}−{x}\right)\right]^{\mathrm{2}} =\frac{\mathrm{log}_{\mathrm{2}} {g}\left({x}\right)}{{x}};{x}\neq\mathrm{0} \\ $$$${f}\left({g}\left({x}\right){x}\right)=\left[{f}\left({g}\left({x}−\mathrm{1}\right){x}\right)\right]^{\mathrm{2}} +\left[{f}\left(\pi/\mathrm{2}−{g}\left({x}−\mathrm{1}\right){x}\right)\right]^{\mathrm{2}} \\ $$$$\mathrm{find}\:{f}\:\mathrm{and}\:{g} \\ $$

Question Number 34803    Answers: 1   Comments: 2

Question Number 34781    Answers: 1   Comments: 0

Question Number 34774    Answers: 1   Comments: 1

find lim_(n→+∞) ((n^p sin^2 (n!))/n^(p+1) ) with0<p<1 .

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{n}^{{p}} \:{sin}^{\mathrm{2}} \left({n}!\right)}{{n}^{{p}+\mathrm{1}} }\:\:{with}\mathrm{0}<{p}<\mathrm{1}\:. \\ $$

Question Number 34771    Answers: 0   Comments: 1

let A(x)= ∫_0 ^1 ln(1+ix^2 )dx find a simple form of f(x) (x∈R)

$${let}\:{A}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{ix}^{\mathrm{2}} \right){dx} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right)\:\:\:\:\left({x}\in{R}\right) \\ $$

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