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Question Number 73227 Answers: 1 Comments: 0

find the polynom T_n wich verify T_n (cosθ)=cos(nθ) ∀n integr ∀θ real 1) find T_0 ,T_1 and T_2 and prove that T_(n+2) =2x T_(n+1) −T_n 3) find deg(T_n ) and T_n (1) ,T_n (−1) 4) find T^′ (cosθ) for 0<θ<π and prove that (1−x^2 )T_n ′′−xT′_n +n^2 T_n =0 5) find roots of T_n and decompose T_n inside R[x] 6) find the value of Π_(k=0) ^(n−1) cos((((2k+1)π)/(2n)))

$${find}\:{the}\:{polynom}\:{T}_{{n}} \:{wich}\:{verify}\:{T}_{{n}} \left({cos}\theta\right)={cos}\left({n}\theta\right) \\ $$$$\forall{n}\:{integr}\:\forall\theta\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{T}_{\mathrm{0}} ,{T}_{\mathrm{1}} {and}\:{T}_{\mathrm{2}} {and}\:{prove}\:{that}\: \\ $$$${T}_{{n}+\mathrm{2}} =\mathrm{2}{x}\:{T}_{{n}+\mathrm{1}} −{T}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{deg}\left({T}_{{n}} \right)\:{and}\:{T}_{{n}} \left(\mathrm{1}\right)\:,{T}_{{n}} \left(−\mathrm{1}\right) \\ $$$$\left.\mathrm{4}\right)\:{find}\:{T}\:^{'} \left({cos}\theta\right)\:{for}\:\mathrm{0}<\theta<\pi\:\:{and}\:{prove}\:{that} \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){T}_{{n}} ''−{xT}'_{{n}} \:+{n}^{\mathrm{2}} \:{T}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{roots}\:{of}\:{T}_{{n}} \:{and}\:{decompose}\:{T}_{{n}} {inside}\:{R}\left[{x}\right] \\ $$$$\left.\mathrm{6}\right)\:{find}\:{the}\:{value}\:{of}\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\left(\frac{\left(\mathrm{2}{k}+\mathrm{1}\right)\pi}{\mathrm{2}{n}}\right) \\ $$

Question Number 73225 Answers: 0 Comments: 0

calculate f(x)=∫_0 ^π ln(x^2 −2xcosθ +1)dθ with x real.

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:{x}\:{real}. \\ $$

Question Number 73224 Answers: 1 Comments: 0

find the coefficient a_k of term x^k in Π_(r=1) ^n (1+x^r ) with 0≤k≤((n(n+1))/2) example: n=100, k=50

$${find}\:{the}\:{coefficient}\:{a}_{{k}} \:{of}\:{term}\:{x}^{{k}} \:{in} \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\mathrm{1}+{x}^{{r}} \right)\:{with}\:\mathrm{0}\leqslant{k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$ \\ $$$${example}:\:{n}=\mathrm{100},\:{k}=\mathrm{50} \\ $$

Question Number 73223 Answers: 0 Comments: 5

lim (0/∞) =^(?) 0 ? lim (∞/0) =^(?) ∞ ?

$$\boldsymbol{{lim}}\:\frac{\mathrm{0}}{\infty}\:\overset{?} {=}\:\mathrm{0}\:? \\ $$$$\boldsymbol{{lim}}\:\frac{\infty}{\mathrm{0}}\:\overset{?} {=}\:\infty\:? \\ $$

Question Number 73222 Answers: 1 Comments: 0

Question Number 73221 Answers: 1 Comments: 0

Question Number 73210 Answers: 1 Comments: 2

Question Number 73203 Answers: 0 Comments: 1

Question Number 73202 Answers: 2 Comments: 3

∫((2x^2 −1+2x(√(x^2 −1)))/(x^2 −x+(x−1)(√(x^2 −1))))dx=? ∫(dx/(x(√(x+1))(√((1−x)^3 ))))=?

$$\int\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}+\mathrm{2}{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{{x}^{\mathrm{2}} −{x}+\left({x}−\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{dx}=? \\ $$$$\int\frac{{dx}}{{x}\sqrt{{x}+\mathrm{1}}\sqrt{\left(\mathrm{1}−{x}\right)^{\mathrm{3}} }}=? \\ $$

Question Number 73200 Answers: 0 Comments: 4

if lim_(x→0^+ ) f(x)=+∞ lim_(x→0^− ) f(x)=+∞ then lim_(x→0) f(x)=+∞ or lim_(x→0) f(x)=not exist

$${if}\:\: \\ $$$$ \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {{lim}f}\left({x}\right)=+\infty \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {{lim}f}\left({x}\right)=+\infty \\ $$$$ \\ $$$${then}\:\underset{{x}\rightarrow\mathrm{0}} {{lim}f}\left({x}\right)=+\infty\:{or}\:\underset{{x}\rightarrow\mathrm{0}} {{lim}f}\left({x}\right)={not}\:{exist} \\ $$

Question Number 73258 Answers: 1 Comments: 0

Question Number 73191 Answers: 1 Comments: 0

find x and y: { ((2x^y −x^(−y) =1)),((log_2 y=(√x))) :}

$${find}\:{x}\:{and}\:{y}: \\ $$$$\begin{cases}{\mathrm{2}{x}^{{y}} −{x}^{−{y}} =\mathrm{1}}\\{{log}_{\mathrm{2}} {y}=\sqrt{{x}}}\end{cases} \\ $$

Question Number 73251 Answers: 1 Comments: 0

Question Number 73188 Answers: 0 Comments: 0

Question Number 73182 Answers: 0 Comments: 3

calculate ∫_0 ^∞ xe^(−x^2 ) arctan(x−(1/x))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{xe}^{−{x}^{\mathrm{2}} } {arctan}\left({x}−\frac{\mathrm{1}}{{x}}\right){dx} \\ $$

Question Number 73181 Answers: 1 Comments: 1

calculate ∫_1 ^(3 ) ((x−2)/(√(x^2 +x+1)))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}\:} \:\frac{{x}−\mathrm{2}}{\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$

Question Number 73180 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((lnx)/((x+1)^3 ))dx

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

Question Number 73179 Answers: 1 Comments: 1

caoculate ∫_0 ^∞ ((arctan(x^2 −1))/(2x^2 +1))dx

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

Question Number 73178 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((ln(2+x^2 ))/(x^2 −x+1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}{dx} \\ $$

Question Number 73160 Answers: 1 Comments: 0

1+tanAtan(A/2)=tanAcot(A/2)−1=secA

$$\mathrm{1}+{tanAtan}\frac{{A}}{\mathrm{2}}={tanAcot}\frac{{A}}{\mathrm{2}}−\mathrm{1}={secA} \\ $$

Question Number 73155 Answers: 0 Comments: 0

reposting a former question... ∫(((x)^(1/5) −1)/((√x)+1))dx= [t=(x)^(1/(10)) → dx=10(x^9 )^(1/(10)) dx] =10∫((t^9 (t−1))/(t^4 −t^3 +t^2 −t+1))dt= =10∫(t^6 −t^4 −t)dt+10∫((t(t^2 −t+1))/(t^4 −t^3 +t^2 −t+1))dt= =((10)/7)t^7 −2t^5 −5t^2 +(5+(√5))∫(t/(t^2 −((1−(√5))/3)t+1))dt+(5−(√5))∫(t/(t^2 −((1+(√5))/2)t+1))dt= and it′s easy to solve these

$$\mathrm{reposting}\:\mathrm{a}\:\mathrm{former}\:\mathrm{question}... \\ $$$$\int\frac{\sqrt[{\mathrm{5}}]{{x}}−\mathrm{1}}{\sqrt{{x}}+\mathrm{1}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt[{\mathrm{10}}]{{x}}\:\rightarrow\:{dx}=\mathrm{10}\sqrt[{\mathrm{10}}]{{x}^{\mathrm{9}} }{dx}\right] \\ $$$$=\mathrm{10}\int\frac{{t}^{\mathrm{9}} \left({t}−\mathrm{1}\right)}{{t}^{\mathrm{4}} −{t}^{\mathrm{3}} +{t}^{\mathrm{2}} −{t}+\mathrm{1}}{dt}= \\ $$$$=\mathrm{10}\int\left({t}^{\mathrm{6}} −{t}^{\mathrm{4}} −{t}\right){dt}+\mathrm{10}\int\frac{{t}\left({t}^{\mathrm{2}} −{t}+\mathrm{1}\right)}{{t}^{\mathrm{4}} −{t}^{\mathrm{3}} +{t}^{\mathrm{2}} −{t}+\mathrm{1}}{dt}= \\ $$$$=\frac{\mathrm{10}}{\mathrm{7}}{t}^{\mathrm{7}} −\mathrm{2}{t}^{\mathrm{5}} −\mathrm{5}{t}^{\mathrm{2}} +\left(\mathrm{5}+\sqrt{\mathrm{5}}\right)\int\frac{{t}}{{t}^{\mathrm{2}} −\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{3}}{t}+\mathrm{1}}{dt}+\left(\mathrm{5}−\sqrt{\mathrm{5}}\right)\int\frac{{t}}{{t}^{\mathrm{2}} −\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}{t}+\mathrm{1}}{dt}= \\ $$$$\mathrm{and}\:\mathrm{it}'\mathrm{s}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{these} \\ $$

Question Number 73147 Answers: 1 Comments: 1

∫((x−6)/(x^3 +1))dx

$$\int\frac{{x}−\mathrm{6}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx} \\ $$$$ \\ $$

Question Number 73144 Answers: 1 Comments: 1

calculte ∫ ((x+(√(2+x^2 )))/(x+1−(√(2+x^2 ))))dx

$${calculte}\:\int\:\:\frac{{x}+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{{x}+\mathrm{1}−\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 73137 Answers: 0 Comments: 8

Question Number 73131 Answers: 0 Comments: 2

solve for x,in terms of: a∈R . x+(√x)+(√(x^2 −a))+(√(x−a^2 ))=a^2

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}:\:\:\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}\:. \\ $$$$\:\:\:\boldsymbol{\mathrm{x}}+\sqrt{\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}}+\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}^{\mathrm{2}} }=\boldsymbol{\mathrm{a}}^{\mathrm{2}} \\ $$

Question Number 73117 Answers: 2 Comments: 1

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