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Question Number 59287    Answers: 1   Comments: 0

A circle x^2 +y^2 −2x−4y−5=0 with centr 0 is cut by a line y=2x+5 at points P and Q. Show that QO is perpendicular to PO.

$$\mathrm{A}\:\mathrm{circle}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{2x}−\mathrm{4y}−\mathrm{5}=\mathrm{0}\:\mathrm{with}\:\mathrm{centr} \\ $$$$\mathrm{0}\:\mathrm{is}\:\mathrm{cut}\:\mathrm{by}\:\mathrm{a}\:\mathrm{line}\:\mathrm{y}=\mathrm{2x}+\mathrm{5}\:\mathrm{at}\:\mathrm{points}\:\mathrm{P}\:\mathrm{and}\:\mathrm{Q}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{QO}\:\mathrm{is}\:\mathrm{perpendicular}\:\mathrm{to}\:\mathrm{PO}. \\ $$

Question Number 59282    Answers: 1   Comments: 1

calculate f(x)=∫_0 ^∞ e^(−x[t]) sin([t])dt with x>0 2) calculate ∫_0 ^∞ e^(−3[t]) sin([t])dt .

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} {sin}\left(\left[{t}\right]\right){dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{3}\left[{t}\right]} {sin}\left(\left[{t}\right]\right){dt}\:. \\ $$

Question Number 59280    Answers: 0   Comments: 0

find Σ_(n=1) ^∞ (([(√(n+1))]−[(√n)])/n^2 )

$${find}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left[\sqrt{{n}+\mathrm{1}}\right]−\left[\sqrt{{n}}\right]}{{n}^{\mathrm{2}} } \\ $$

Question Number 59279    Answers: 0   Comments: 5

find ∫_0 ^(π/2) (x/(sinx))dx

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{x}}{{sinx}}{dx} \\ $$

Question Number 59278    Answers: 0   Comments: 3

let f(x) =∫_0 ^∞ ((ln(1+xt^2 ))/(2+t^2 )) dt determine a explicit form of f(x) 2)calculate ∫_0 ^∞ ((ln(1+3x^2 ))/(2+t^2 ))dt

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }\:{dt} \\ $$$${determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 59277    Answers: 0   Comments: 1

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

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

Question Number 59276    Answers: 0   Comments: 0

find the value of Σ_(n=2) ^∞ (n/((n−1)^3 (n+1)^3 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\frac{{n}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 59275    Answers: 0   Comments: 3

let f(x)=∫_0 ^1 ln(1+xe^(−t) )dt 1)find a explicit form of f(x) 2)calculate ∫_0 ^1 ln(1+2e^(−t) )dt 3) developp f(x)at integr serie if ∣x∣<1

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xe}^{−{t}} \right){dt} \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+\mathrm{2}{e}^{−{t}} \right){dt} \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\left({x}\right){at}\:{integr}\:{serie}\:{if}\:\mid{x}\mid<\mathrm{1} \\ $$

Question Number 59274    Answers: 1   Comments: 2

calculate ∫_0 ^1 arctan(1+x+x^2 )dx

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

Question Number 59273    Answers: 1   Comments: 0

abc = 64 a, b, c ∈ R^+ Find K that satisfy to the inequality : (((a + b) (√(ab)) + (b + c) (√(bc)) + (c + a) (√(ca)))/(√(abc))) ≥ (√a) + (√b) + (√c) + K .

$${abc}\:\:=\:\:\mathrm{64} \\ $$$${a},\:{b},\:{c}\:\:\in\:\:\mathbb{R}^{+} \\ $$$${Find}\:\:{K}\:\:{that}\:\:{satisfy}\:\:{to}\:\:{the}\:\:{inequality}\:\:: \\ $$$$\:\:\:\:\:\frac{\left({a}\:+\:{b}\right)\:\sqrt{{ab}}\:\:+\:\:\left({b}\:+\:{c}\right)\:\sqrt{{bc}}\:\:+\:\:\left({c}\:+\:{a}\right)\:\sqrt{{ca}}}{\sqrt{{abc}}}\:\:\:\geqslant\:\:\sqrt{{a}}\:\:+\:\:\sqrt{{b}}\:\:+\:\:\sqrt{{c}}\:\:+\:\:{K}\:\:. \\ $$

Question Number 59270    Answers: 1   Comments: 0

Solve the equation 3x^(1/2) +5−2x^(1/2) =0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{3x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{5}−\mathrm{2x}^{\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{0} \\ $$

Question Number 59342    Answers: 1   Comments: 0

Question Number 59260    Answers: 1   Comments: 0

Question Number 59247    Answers: 1   Comments: 2

let f(x) =∫_0 ^(π/2) ln(1−xcost)dt with ∣x∣<1 1) developp f at integr serie 2) find a explicit form of f(x) 3) find the values of integrals ∫_0 ^(π/2) ln(1−cost)dt and ∫_0 ^(π/2) ln(1+cost)dt 4) calculate U_n =∫_0 ^(π/2) ln(1−(2/n)cost)dt with n integr and n≥2 study the convergence of U_n and Σ U_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−{xcost}\right){dt}\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−{cost}\right){dt}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{cost}\right){dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−\frac{\mathrm{2}}{{n}}{cost}\right){dt}\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$$${study}\:{the}\:{convergence}\:{of}\:{U}_{{n}} \:\:\:\:{and}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 59246    Answers: 3   Comments: 4

Question Number 59225    Answers: 1   Comments: 3

Question Number 59226    Answers: 2   Comments: 0

Question Number 59220    Answers: 2   Comments: 0

find the minimum value of y= (b^2 /(sin^2 x))+(a^2 /(cos^2 x))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{y}=\:\frac{\mathrm{b}^{\mathrm{2}} }{\mathrm{sin}^{\mathrm{2}} \mathrm{x}}+\frac{\mathrm{a}^{\mathrm{2}} }{\mathrm{cos}^{\mathrm{2}} \mathrm{x}} \\ $$

Question Number 59214    Answers: 0   Comments: 0

Question Number 59212    Answers: 0   Comments: 1

Question Number 59201    Answers: 2   Comments: 4

make x the subject of x=m+x

$$\mathrm{make}\:\mathrm{x}\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\:\mathrm{x}=\mathrm{m}+\mathrm{x} \\ $$

Question Number 59199    Answers: 0   Comments: 4

Question Number 59193    Answers: 1   Comments: 0

Question Number 59190    Answers: 0   Comments: 0

find ∫_0 ^1 (x^2 /(1−cosx))dx .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}−{cosx}}{dx}\:. \\ $$

Question Number 59188    Answers: 2   Comments: 4

let f(x)=x−(√(4−x^2 )) and g(x) =((2 +(√(x−3)))/(2−(√(x−3)))) 1) find D_f ,D_g and D_(fog) and determine fog(x) 2) calculate gof(x) and give D_(gof) 3) calculate ∫_(−(1/2)) ^(1/2) f(x)dx 4) calculate ∫_4 ^5 g(x)dx .

$${let}\:{f}\left({x}\right)={x}−\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:\:{and}\:{g}\left({x}\right)\:=\frac{\mathrm{2}\:+\sqrt{{x}−\mathrm{3}}}{\mathrm{2}−\sqrt{{x}−\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:\:{D}_{{f}} \:\:,{D}_{{g}} \:\:\:{and}\:{D}_{{fog}} \:\:\:\:\:{and}\:\:{determine}\:{fog}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{gof}\left({x}\right)\:{and}\:{give}\:{D}_{{gof}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} {f}\left({x}\right){dx}\:\:\:\: \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:\int_{\mathrm{4}} ^{\mathrm{5}} \:{g}\left({x}\right){dx}\:. \\ $$

Question Number 59187    Answers: 0   Comments: 2

calculate lim_(x→0) ((ln(arctan(1+x))−ln((π/4)))/x^2 )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left({arctan}\left(\mathrm{1}+{x}\right)\right)−{ln}\left(\frac{\pi}{\mathrm{4}}\right)}{{x}^{\mathrm{2}} } \\ $$

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