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Question Number 39520    Answers: 1   Comments: 1

if (1+x)^n =Σ_(i=0) ^n a_i x^i and (1+x)^(n+1) =Σ_(i=0) ^(n+1) b_i x^i calculate ((∐_(i=0) ^n a_i )/(Π_(i=0) ^(n+1) b_i )) .

$${if}\:\left(\mathrm{1}+{x}\right)^{{n}} \:=\sum_{{i}=\mathrm{0}} ^{{n}} \:{a}_{{i}} {x}^{{i}} \:\:\:\:{and} \\ $$$$\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} \:=\sum_{{i}=\mathrm{0}} ^{{n}+\mathrm{1}} \:{b}_{{i}} \:{x}^{{i}} \:\:{calculate} \\ $$$$\frac{\coprod_{{i}=\mathrm{0}} ^{{n}} \:{a}_{{i}} }{\prod_{{i}=\mathrm{0}} ^{{n}+\mathrm{1}} \:{b}_{{i}} }\:. \\ $$

Question Number 39519    Answers: 1   Comments: 1

simplify 1) A_n =(1/(√a)){ (((1+(√a))/2) )^n −(((1−(√a))/2))^n } with n natural integr and a>0 2) f(x)= (1/(√(2x+1))){ (((1+(√(2x+1)))/2))^n −(((1−(√(2x+1)))/2))^n }

$${simplify}\: \\ $$$$\left.\mathrm{1}\right)\:{A}_{{n}} =\frac{\mathrm{1}}{\sqrt{{a}}}\left\{\:\left(\frac{\mathrm{1}+\sqrt{{a}}}{\mathrm{2}}\:\right)^{{n}} \:−\left(\frac{\mathrm{1}−\sqrt{{a}}}{\mathrm{2}}\right)^{{n}} \right\}\:{with}\:{n}\:{natural}\: \\ $$$${integr}\:{and}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{x}+\mathrm{1}}}\left\{\:\left(\frac{\mathrm{1}+\sqrt{\mathrm{2}{x}+\mathrm{1}}}{\mathrm{2}}\right)^{{n}} \:−\left(\frac{\mathrm{1}−\sqrt{\mathrm{2}{x}+\mathrm{1}}}{\mathrm{2}}\right)^{{n}} \right\} \\ $$

Question Number 39517    Answers: 0   Comments: 2

find radius of S(x)=Σ_(n=1) ^∞ (x^n /n^2 ) and calculate its sum 2) find Σ_(n=1) ^∞ (1/n^2 ) and Σ_(n=1) ^∞ (1/(n^2 2^n )) .

$${find}\:{radius}\:{of}\:\:{S}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{x}^{{n}} }{{n}^{\mathrm{2}} } \\ $$$${and}\:{calculate}\:{its}\:{sum} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:\:\:{and}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:\mathrm{2}^{{n}} }\:. \\ $$$$ \\ $$

Question Number 39524    Answers: 0   Comments: 0

Question Number 39511    Answers: 0   Comments: 2

Question Number 39508    Answers: 1   Comments: 1

Question Number 39486    Answers: 0   Comments: 7

sin θ=sin αsin (((θ+α)/2)) Express θ explicitly in terms of α.

$$\mathrm{sin}\:\theta=\mathrm{sin}\:\alpha\mathrm{sin}\:\left(\frac{\theta+\alpha}{\mathrm{2}}\right) \\ $$$${Express}\:\theta\:{explicitly}\:{in}\:{terms}\:{of}\:\alpha. \\ $$

Question Number 39475    Answers: 1   Comments: 0

Question Number 39470    Answers: 2   Comments: 2

Question Number 39469    Answers: 1   Comments: 0

Question Number 39466    Answers: 0   Comments: 1

Question Number 39464    Answers: 1   Comments: 0

Domain of the explicit form of the function y represented implicitly by the equation (1+x)cosy−x^2 =0 is (a) (−1,1] (b) (−1, 1−(√)5/2] (c) [1−(√)5/2, 1+(√)5/2] (d) [0, 1+(√)5/2]

$${Domain}\:\:{of}\:\:{the}\:\:{explicit}\:\:{form}\:\:{of} \\ $$$${the}\:\:{function}\:\:\:{y}\:\:\:{represented}\: \\ $$$${implicitly}\:\:\:{by}\:\:{the}\:\:{equation}\: \\ $$$$\left(\mathrm{1}+{x}\right){cosy}−{x}^{\mathrm{2}} =\mathrm{0}\:\:{is} \\ $$$$\left({a}\right)\:\:\left(−\mathrm{1},\mathrm{1}\right]\:\:\:\:\:\:\:\:\:\:\left({b}\right)\:\:\:\:\left(−\mathrm{1},\:\mathrm{1}−\sqrt{}\mathrm{5}/\mathrm{2}\right] \\ $$$$\left({c}\right)\:\:\:\left[\mathrm{1}−\sqrt{}\mathrm{5}/\mathrm{2},\:\mathrm{1}+\sqrt{}\mathrm{5}/\mathrm{2}\right] \\ $$$$\left({d}\right)\:\:\left[\mathrm{0},\:\mathrm{1}+\sqrt{}\mathrm{5}/\mathrm{2}\right] \\ $$

Question Number 39458    Answers: 0   Comments: 4

find the greatest possible square insribed in a triangle with sides a b c

$$\mathrm{find}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{possible}\:\mathrm{square}\:\mathrm{insribed}\:\mathrm{in} \\ $$$$\mathrm{a}\:\mathrm{triangle}\:\mathrm{with}\:\mathrm{sides}\:{a}\:{b}\:{c} \\ $$

Question Number 39457    Answers: 1   Comments: 0

Question Number 39443    Answers: 1   Comments: 3

lim_(n→∞) [ (1/(n^2 +1))+ (2/(n^2 +2))+ (3/(n^2 +3))+ ....+(1/(n+1))] = ?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left[\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}}+\:\frac{\mathrm{2}}{\mathrm{n}^{\mathrm{2}} +\mathrm{2}}+\:\frac{\mathrm{3}}{\mathrm{n}^{\mathrm{2}} +\mathrm{3}}+\:....+\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right]\:=\:? \\ $$

Question Number 39441    Answers: 0   Comments: 2

∫_(1/4) ^( 4) (1/x) sin (x−(1/x))dx = ?

$$\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\:\mathrm{4}} \:\frac{\mathrm{1}}{{x}}\:\mathrm{sin}\:\left({x}−\frac{\mathrm{1}}{{x}}\right){dx}\:=\:? \\ $$

Question Number 39440    Answers: 1   Comments: 0

f(x)= ∫_0 ^( x_ ) e^(t ) (((1+sin t)/(1+cos t))) dt. Then f((π/3))×f(((2π)/3)) = ?

$$\mathrm{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\:{x}_{} } \:{e}^{{t}\:} \left(\frac{\mathrm{1}+\mathrm{sin}\:{t}}{\mathrm{1}+\mathrm{cos}\:{t}}\right)\:{dt}. \\ $$$${T}\mathrm{hen}\:\:\mathrm{f}\left(\frac{\pi}{\mathrm{3}}\right)×\mathrm{f}\left(\frac{\mathrm{2}\pi}{\mathrm{3}}\right)\:=\:? \\ $$

Question Number 39436    Answers: 1   Comments: 2

Question Number 39477    Answers: 1   Comments: 3

∫2^x 3^(2x) dx=?

$$\int\mathrm{2}^{\mathrm{x}} \mathrm{3}^{\mathrm{2x}} \mathrm{dx}=? \\ $$

Question Number 39431    Answers: 1   Comments: 1

∫_0 ^(2π) e^(x/2) sin ((x/2)+(π/4))dx = ?

$$\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\mathrm{e}^{\frac{{x}}{\mathrm{2}}} \mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right)\mathrm{d}{x}\:=\:? \\ $$

Question Number 39402    Answers: 1   Comments: 1

Question Number 39483    Answers: 0   Comments: 3

find f(t)= ∫_0 ^1 ((ln(1+xt))/(1+x^2 )) dx .

$${find}\:{f}\left({t}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{xt}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 39395    Answers: 0   Comments: 2

Given that θ is an obtuse angle find tan θ if cos θ =(3/5)

$${Given}\:{that}\:\theta\:{is}\:{an}\:{obtuse}\: \\ $$$${angle}\:{find}\:{tan}\:\theta\:{if} \\ $$$${cos}\:\theta\:=\frac{\mathrm{3}}{\mathrm{5}} \\ $$$$ \\ $$

Question Number 39389    Answers: 0   Comments: 2

calculate F(x) = ∫_0 ^∞ (dt/(1+(1+x(1+t^2 ))^2 ))

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

Question Number 39388    Answers: 1   Comments: 0

calculate A =tan((π/5)).tan(((2π)/5)).tan(((3π)/5)).tan(((4π)/5))

$${calculate}\:{A}\:={tan}\left(\frac{\pi}{\mathrm{5}}\right).{tan}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right).{tan}\left(\frac{\mathrm{3}\pi}{\mathrm{5}}\right).{tan}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right) \\ $$

Question Number 39386    Answers: 1   Comments: 1

find the value of ∫_0 ^1 ((ln(1+x))/(1+x^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

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