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Question Number 101974    Answers: 2   Comments: 1

Question Number 101971    Answers: 0   Comments: 2

Question Number 101970    Answers: 1   Comments: 0

∫_0 ^∞ ((Cos(ax))/(x^2 +b^2 )) dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{Cos}\left({ax}\right)}{{x}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 101961    Answers: 0   Comments: 4

Question Number 101959    Answers: 1   Comments: 0

Starting from y=(4/(√π))t^((3/2) ) ∫_0 ^∞ x^3 e^(−tx^2 ) dx find (π/8)=?

$$ \\ $$$$ \\ $$$${Starting}\:{from} \\ $$$$\:\:\:\:\:\:\:\:{y}=\frac{\mathrm{4}}{\sqrt{\pi}}{t}^{\frac{\mathrm{3}}{\mathrm{2}}\:} \int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{3}} {e}^{−{tx}^{\mathrm{2}} } {dx} \\ $$$${find}\:\:\:\frac{\pi}{\mathrm{8}}=? \\ $$

Question Number 101951    Answers: 0   Comments: 3

Question Number 101940    Answers: 1   Comments: 0

lim_(n→∞ ) (1/n^2 )(ne^((−1)/n^2 ) +ne^((−4)/n^2 ) +.....∞)

$$\underset{{n}\rightarrow\infty\:} {\mathrm{lim}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left({ne}^{\frac{−\mathrm{1}}{{n}^{\mathrm{2}} }} +{ne}^{\frac{−\mathrm{4}}{{n}^{\mathrm{2}} }} +.....\infty\right) \\ $$

Question Number 101937    Answers: 0   Comments: 0

Given a circle with the center at the point O and the radius of the length R.From a point A outside so that AO=2R,drawing two tangents AB and AC to the circle (B and C are the tangency points).Take a arbitrary point M on smaller arc BC (M differ from B and C) The tangent pass M cuts AB and AC at Pand Q respectively.The segments OP and OQ cuts BC at D and E respectively. i)Prove that PQ=2DE ii)Define the position of M such the area of the triangle ODE is smallest and expression it by R

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{the}\:\mathrm{center}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{O}\: \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{length}\:\mathrm{R}.\mathrm{From}\:\mathrm{a}\:\mathrm{point}\:\mathrm{A}\:\mathrm{outside} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{AO}=\mathrm{2R},\mathrm{drawing}\:\mathrm{two}\:\mathrm{tangents}\:\mathrm{AB}\:\mathrm{and}\:\mathrm{AC}\:\mathrm{to}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\left(\mathrm{B}\:\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{the}\:\mathrm{tangency}\:\mathrm{points}\right).\mathrm{Take}\:\mathrm{a}\:\mathrm{arbitrary}\:\mathrm{point}\:\mathrm{M} \\ $$$$\mathrm{on}\:\mathrm{smaller}\:\mathrm{arc}\:\mathrm{BC}\:\left(\mathrm{M}\:\mathrm{differ}\:\mathrm{from}\:\mathrm{B}\:\mathrm{and}\:\mathrm{C}\right) \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{pass}\:\mathrm{M}\:\mathrm{cuts}\:\mathrm{AB}\:\mathrm{and}\:\mathrm{AC}\:\mathrm{at}\:\mathrm{Pand}\:\mathrm{Q} \\ $$$$\mathrm{respectively}.\mathrm{The}\:\mathrm{segments}\:\mathrm{OP}\:\mathrm{and}\:\mathrm{OQ}\:\mathrm{cuts} \\ $$$$\mathrm{BC}\:\mathrm{at}\:\mathrm{D}\:\mathrm{and}\:\mathrm{E}\:\mathrm{respectively}. \\ $$$$\left.\mathrm{i}\right)\mathrm{Prove}\:\mathrm{that}\:\mathrm{PQ}=\mathrm{2DE} \\ $$$$\left.\mathrm{ii}\right)\mathrm{Define}\:\:\mathrm{the}\:\mathrm{position}\:\mathrm{of}\:\mathrm{M}\:\mathrm{such}\:\mathrm{the}\: \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{ODE}\:\mathrm{is}\:\mathrm{smallest} \\ $$$$\mathrm{and}\:\mathrm{expression}\:\mathrm{it}\:\mathrm{by}\:\mathrm{R} \\ $$

Question Number 101921    Answers: 1   Comments: 1

Please, help i cannot solve problems such as: x^4 =1, x^5 =1, or x^n =1 n∈N...

$$\boldsymbol{{Please}},\:\boldsymbol{{help}}\:\:\boldsymbol{{i}}\:\:\boldsymbol{{cannot}}\:\boldsymbol{{solve}}\:\boldsymbol{{problems}} \\ $$$$\boldsymbol{{such}}\:\boldsymbol{{as}}:\:\boldsymbol{{x}}^{\mathrm{4}} =\mathrm{1},\:\boldsymbol{{x}}^{\mathrm{5}} =\mathrm{1},\:\boldsymbol{{or}}\:\boldsymbol{{x}}^{\boldsymbol{{n}}} =\mathrm{1}\:\boldsymbol{{n}}\in\mathbb{N}... \\ $$

Question Number 101891    Answers: 3   Comments: 0

if x a integer number , when divided 8 has remainder 5 and divided 5 has remainder 2. find x

$${if}\:{x}\:{a}\:{integer}\:{number}\:,\:{when}\:{divided}\:\mathrm{8} \\ $$$${has}\:{remainder}\:\mathrm{5}\:{and}\:{divided}\:\mathrm{5}\:{has}\:{remainder} \\ $$$$\mathrm{2}.\:{find}\:{x} \\ $$

Question Number 101888    Answers: 3   Comments: 1

If x an integer when divided 5 give remainder 2 and when divided 4 give remainder 3. find the value of x

$${If}\:{x}\:{an}\:{integer}\:{when}\:{divided} \\ $$$$\mathrm{5}\:{give}\:{remainder}\:\mathrm{2}\:{and}\:{when}\:{divided} \\ $$$$\mathrm{4}\:{give}\:{remainder}\:\mathrm{3}. \\ $$$${find}\:{the}\:{value}\:{of}\:{x} \\ $$

Question Number 102323    Answers: 2   Comments: 0

Solve this linear Equation (dy/dx) + y cos x = (1/2) sin x

$${Solve}\:{this}\:{linear}\:{Equation} \\ $$$$\frac{{dy}}{{dx}}\:+\:{y}\:{cos}\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:{sin}\:{x} \\ $$

Question Number 102313    Answers: 2   Comments: 0

Question Number 101882    Answers: 1   Comments: 1

∫(1/((√x)+(√(x+1))+(√(x+2))))dx

$$\:\:\:\int\frac{\mathrm{1}}{\sqrt{\boldsymbol{{x}}}+\sqrt{\boldsymbol{{x}}+\mathrm{1}}+\sqrt{\boldsymbol{{x}}+\mathrm{2}}}\boldsymbol{{dx}} \\ $$

Question Number 101880    Answers: 2   Comments: 0

Find the number of six−digit odd numbers without repeated digits.

$${Find}\:{the}\:{number}\:{of}\:{six}−{digit}\:{odd} \\ $$$${numbers}\:{without}\:{repeated}\:{digits}. \\ $$

Question Number 105257    Answers: 2   Comments: 0

y′′−2y′+y = xe^x sin x

$${y}''−\mathrm{2}{y}'+{y}\:=\:{xe}^{{x}} \mathrm{sin}\:{x}\: \\ $$

Question Number 105256    Answers: 0   Comments: 0

Question Number 101848    Answers: 2   Comments: 0

(cos x) (dy/dx) +y sin x = 2x cos^2 x , y((π/4)) = ((−15π^2 (√2))/(32))

$$\left(\mathrm{cos}\:\mathrm{x}\right)\:\frac{\mathrm{dy}}{\mathrm{dx}}\:+\mathrm{y}\:\mathrm{sin}\:\mathrm{x}\:=\:\mathrm{2x}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:, \\ $$$$\mathrm{y}\left(\frac{\pi}{\mathrm{4}}\right)\:=\:\frac{−\mathrm{15}\pi^{\mathrm{2}} \sqrt{\mathrm{2}}}{\mathrm{32}} \\ $$

Question Number 101846    Answers: 2   Comments: 0

Question Number 101841    Answers: 1   Comments: 0

xy′ + y = y^2

$${xy}'\:+\:{y}\:=\:{y}^{\mathrm{2}} \\ $$

Question Number 101835    Answers: 3   Comments: 0

∫_0 ^∞ (1/(e^x +1)) dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{1}}{{e}^{{x}} +\mathrm{1}}\:{dx}\: \\ $$

Question Number 101833    Answers: 3   Comments: 0

∫ _(−1)^1 (√((1+x)/(1−x))) dx ?

$$\int\:_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}\:{dx}\:?\: \\ $$

Question Number 101832    Answers: 0   Comments: 1

∫((ln x)/(x^2 +1)) dx ? (JS ⊛)

$$\int\frac{\mathrm{ln}\:{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}\:?\: \\ $$$$\left({JS}\:\circledast\right) \\ $$

Question Number 101828    Answers: 2   Comments: 0

∫_0 ^∞ ((e^(πx) −e^x )/(x(e^(πx) +1)(e^x +1)))dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{e}^{\pi\mathrm{x}} −\mathrm{e}^{\mathrm{x}} }{\mathrm{x}\left(\mathrm{e}^{\pi\mathrm{x}} +\mathrm{1}\right)\left(\mathrm{e}^{\mathrm{x}} +\mathrm{1}\right)}\mathrm{dx} \\ $$

Question Number 101822    Answers: 1   Comments: 0

lim_(n→∞) ((φ^(n+1) −(−φ)^(−n−1) )/(φ^n −(−φ)^(−n) )) = (JS ⊛)

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\phi^{{n}+\mathrm{1}} −\left(−\phi\right)^{−{n}−\mathrm{1}} }{\phi^{{n}} −\left(−\phi\right)^{−{n}} }\:=\: \\ $$$$\left({JS}\:\circledast\right) \\ $$

Question Number 101816    Answers: 1   Comments: 0

∫_1 ^( e) (((tan^(−1) x)/x)+((log)/(x^2 +1)))dx

$$\int_{\mathrm{1}} ^{\:\:{e}} \left(\frac{{tan}^{−\mathrm{1}} {x}}{{x}}+\frac{{log}}{{x}^{\mathrm{2}} +\mathrm{1}}\right){dx} \\ $$

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