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Question Number 109307 Answers: 1 Comments: 0

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Question Number 109302 Answers: 5 Comments: 0

Question Number 109290 Answers: 2 Comments: 0

Question Number 109284 Answers: 7 Comments: 0

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Question Number 109337 Answers: 2 Comments: 0

x=cosθ, where ((3π)/2)<θ<2π, and that 2cosθ−sinθ=2, show that (√(1−x^2 ))=2(1−x). Hence or otherwise, find x and deduce that tan2θ=((24)/7)

$${x}=\mathrm{cos}\theta,\:\mathrm{where}\:\frac{\mathrm{3}\pi}{\mathrm{2}}<\theta<\mathrm{2}\pi,\:\mathrm{and}\:\mathrm{that}\:\mathrm{2cos}\theta−\mathrm{sin}\theta=\mathrm{2}, \\ $$$$\mathrm{show}\:\mathrm{that}\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }=\mathrm{2}\left(\mathrm{1}−{x}\right). \\ $$$$\mathrm{Hence}\:\mathrm{or}\:\mathrm{otherwise},\:\mathrm{find}\:{x}\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that}\:\mathrm{tan2}\theta=\frac{\mathrm{24}}{\mathrm{7}} \\ $$

Question Number 109264 Answers: 1 Comments: 0

Question Number 109262 Answers: 2 Comments: 0

f(x)+f(2x+y)+5xy = f(3x−y)+x^2 +1 for every x,y∈R . find f(10)

$${f}\left({x}\right)+{f}\left(\mathrm{2}{x}+{y}\right)+\mathrm{5}{xy}\:=\:{f}\left(\mathrm{3}{x}−{y}\right)+{x}^{\mathrm{2}} +\mathrm{1} \\ $$$${for}\:{every}\:{x},{y}\in\mathbb{R}\:.\:{find}\:{f}\left(\mathrm{10}\right) \\ $$

Question Number 109248 Answers: 0 Comments: 0

Question Number 109246 Answers: 5 Comments: 0

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Question Number 109240 Answers: 1 Comments: 0

((♭emath)/(•••••)) use cayley − hamilton theorem to calculate A^(−1) for A= (((1 2 2)),((1 2 −1)),((−1 1 4)) )

$$\:\:\frac{\flat{emath}}{\bullet\bullet\bullet\bullet\bullet} \\ $$$${use}\:{cayley}\:−\:{hamilton}\:{theorem} \\ $$$${to}\:{calculate}\:{A}^{−\mathrm{1}} \:{for}\:{A}=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\mathrm{2}\:\:\:\:\:\mathrm{2}}\\{\mathrm{1}\:\:\:\:\:\:\mathrm{2}\:\:−\mathrm{1}}\\{−\mathrm{1}\:\:\mathrm{1}\:\:\:\:\mathrm{4}}\end{pmatrix} \\ $$

Question Number 109222 Answers: 1 Comments: 0

((…♭emATH…)/(≅≅≅≅≅≅)) (√(5+(√(10)))) = x.((√(5+(√(15)) )) +(√(5−(√(15)))) ) x =?

$$\:\:\:\:\frac{\ldots\flat{em}\mathcal{ATH}\ldots}{\cong\cong\cong\cong\cong\cong} \\ $$$$\sqrt{\mathrm{5}+\sqrt{\mathrm{10}}}\:=\:{x}.\left(\sqrt{\mathrm{5}+\sqrt{\mathrm{15}}\:}\:+\sqrt{\mathrm{5}−\sqrt{\mathrm{15}}}\:\right) \\ $$$${x}\:=? \\ $$

Question Number 109220 Answers: 0 Comments: 0

calculate I_n =∫_0 ^(2π) ((cos(nx))/(cosx +sinx))dx (n→natural)

$$\mathrm{calculate}\:\:\mathrm{I}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{\mathrm{cos}\left(\mathrm{nx}\right)}{\mathrm{cosx}\:+\mathrm{sinx}}\mathrm{dx}\:\:\left(\mathrm{n}\rightarrow\mathrm{natural}\right) \\ $$

Question Number 109219 Answers: 0 Comments: 0

let f(x) =((sin(αx))/(sinx)) , 2π periodi even developp f at fourier serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{sin}\left(\alpha\mathrm{x}\right)}{\mathrm{sinx}}\:\:\:\:\:,\:\mathrm{2}\pi\:\mathrm{periodi}\:\mathrm{even} \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 109218 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(2+2t^2 ))/(1+t^2 ))dt

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

Question Number 109217 Answers: 0 Comments: 0

find U_n =∫_0 ^1 sin(narctanx)dx

$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{sin}\left(\mathrm{narctanx}\right)\mathrm{dx} \\ $$

Question Number 109215 Answers: 0 Comments: 1

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

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

Question Number 109214 Answers: 1 Comments: 0

calculateA_n = ∫_0 ^∞ (dx/((x^2 +n)(x^2 +2n))) with n integr natural≥1

$$\mathrm{calculateA}_{\mathrm{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{n}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{2n}\right)}\:\:\mathrm{with}\:\mathrm{n}\:\mathrm{integr}\:\mathrm{natural}\geqslant\mathrm{1} \\ $$

Question Number 109213 Answers: 0 Comments: 0

find nature of Σ_(n=0) ^∞ (((−1)^([x]) )/(2+cos(n[x]))) with [..] mean floor

$$\mathrm{find}\:\mathrm{nature}\:\mathrm{of}\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\left[\mathrm{x}\right]} }{\mathrm{2}+\mathrm{cos}\left(\mathrm{n}\left[\mathrm{x}\right]\right)}\:\:\mathrm{with}\:\left[..\right]\:\mathrm{mean}\:\mathrm{floor} \\ $$

Question Number 109212 Answers: 0 Comments: 0

calculate ∫_0 ^π ((sin(nx))/(cosx))dx with n integr

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{sin}\left(\mathrm{nx}\right)}{\mathrm{cosx}}\mathrm{dx}\:\:\mathrm{with}\:\mathrm{n}\:\mathrm{integr} \\ $$

Question Number 109208 Answers: 1 Comments: 0

∫_0 ^∞ ((x^n −1)/(x−1)).(x/e^x )dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{n}} −\mathrm{1}}{{x}−\mathrm{1}}.\frac{{x}}{{e}^{{x}} }{dx} \\ $$

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