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Question Number 116601 Answers: 2 Comments: 0

If (x,y) satisfies the system of equations { ((∣x∣−x−y+2=0)),((∣y∣+y+5x=1)) :} find the value of x+y.

$$\mathrm{If}\:\left({x},{y}\right)\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\begin{cases}{\mid{x}\mid−{x}−{y}+\mathrm{2}=\mathrm{0}}\\{\mid{y}\mid+{y}+\mathrm{5}{x}=\mathrm{1}}\end{cases}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}+{y}.\: \\ $$

Question Number 116595 Answers: 1 Comments: 0

If (1−2x+3x^2 )^(10) =a_0 +a_1 x+a_2 x^2 +...+a_(20) x^(20) find the value of a_1 +a_2 +...+a_(20)

$$\mathrm{If}\:\left(\mathrm{1}−\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{10}} ={a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +...+{a}_{\mathrm{20}} {x}^{\mathrm{20}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{\mathrm{20}} \\ $$

Question Number 116590 Answers: 2 Comments: 0

∫ ((√x)/(1+x^3 )) dx =?

$$\:\int\:\frac{\sqrt{\mathrm{x}}}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx}\:=? \\ $$

Question Number 116588 Answers: 2 Comments: 0

lim_(x→0) ((4x)/(2 cosec x (1−(√(cos x))))) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{4x}}{\mathrm{2}\:\mathrm{cosec}\:\mathrm{x}\:\left(\mathrm{1}−\sqrt{\mathrm{cos}\:\mathrm{x}}\right)}\:=? \\ $$

Question Number 116586 Answers: 0 Comments: 0

1) explicite ∫_0 ^∞ ((arctan(1+x(2+t^2 )))/(2+t^2 ))dt withx>0 2)determine values of ∫_0 ^∞ ((arctan(3+t^2 ))/(2+t^2 ))dt and ∫_0 ^∞ ((arctan(5+2t^2 ))/(2+t^2 ))dt

$$\left.\mathrm{1}\right)\:{explicite}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{1}+{x}\left(\mathrm{2}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$$${withx}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){determine}\:{values}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{3}+{t}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt}\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left(\mathrm{5}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 116578 Answers: 3 Comments: 0

solve for x determinant (((1 x x^2 x^3 )),((1 2 2^2 2^3 )),((1 3 3^2 3^3 )),((1 4 4^2 4^3 )))= 0

$$\mathrm{solve}\:\mathrm{for}\:\mathrm{x}\: \\ $$$$\:\begin{vmatrix}{\mathrm{1}\:\:\:\mathrm{x}\:\:\:\:\mathrm{x}^{\mathrm{2}} \:\:\:\:\mathrm{x}^{\mathrm{3}} }\\{\mathrm{1}\:\:\:\mathrm{2}\:\:\:\:\mathrm{2}^{\mathrm{2}} \:\:\:\:\mathrm{2}^{\mathrm{3}} }\\{\mathrm{1}\:\:\:\mathrm{3}\:\:\:\:\mathrm{3}^{\mathrm{2}} \:\:\:\:\mathrm{3}^{\mathrm{3}} }\\{\mathrm{1}\:\:\:\mathrm{4}\:\:\:\:\mathrm{4}^{\mathrm{2}} \:\:\:\:\mathrm{4}^{\mathrm{3}} }\end{vmatrix}=\:\mathrm{0} \\ $$

Question Number 116576 Answers: 3 Comments: 1

lim_(x→0) (((sin x)/x))^(1/x^2 ) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\right)^{\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }} \:=? \\ $$

Question Number 116564 Answers: 2 Comments: 0

∫ (dx/(x+(x)^(1/(3 )) )) ?

$$\int\:\frac{\mathrm{dx}}{\mathrm{x}+\sqrt[{\mathrm{3}\:}]{\mathrm{x}}}\:? \\ $$

Question Number 116560 Answers: 1 Comments: 1

if arctan(x+iy) =a+ib with a and b reals determine a and b

$${if}\:{arctan}\left({x}+{iy}\right)\:={a}+{ib} \\ $$$${with}\:{a}\:{and}\:{b}\:{reals}\:{determine} \\ $$$${a}\:{and}\:{b} \\ $$

Question Number 116557 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((ln(1+x(1+t^2 ))/(1+t^2 )) dt with x>0 2) find the value of ∫_0 ^∞ ((ln(2+t^2 ))/(1+t^2 ))dt and ∫_0 ^∞ ((ln(3+2t^2 ))/(1+t^2 ))dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right.}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$${with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{2}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$${and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{3}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 116556 Answers: 1 Comments: 0

let g(x)=ln(cos(ax)) developp g at fourier serie (a real given)

$${let}\:{g}\left({x}\right)={ln}\left({cos}\left({ax}\right)\right) \\ $$$${developp}\:{g}\:{at}\:{fourier}\:{serie} \\ $$$$\left({a}\:{real}\:{given}\right) \\ $$

Question Number 116555 Answers: 1 Comments: 0

find Σ_(n=1) ^∞ (((−1)^n )/(n^2 (n+1)^3 (n+2)^4 ))

$${find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{2}\right)^{\mathrm{4}} } \\ $$

Question Number 116554 Answers: 1 Comments: 0

study the integral ∫_0 ^∞ ((sin^n x)/x^n )dx n integr and n≥2

$${study}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{{n}} {x}}{{x}^{{n}} }{dx} \\ $$$${n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2} \\ $$

Question Number 116538 Answers: 0 Comments: 2

x+(√y)=11 (√x)−y=24 x,y=?

$$\mathrm{x}+\sqrt{\mathrm{y}}=\mathrm{11} \\ $$$$\sqrt{\mathrm{x}}−\mathrm{y}=\mathrm{24} \\ $$$$\mathrm{x},\mathrm{y}=? \\ $$

Question Number 116533 Answers: 2 Comments: 0

find u_n =∫_0 ^∞ ((sin^n x)/x)dx (n≥1)

$${find}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}^{{n}} {x}}{{x}}{dx}\:\:\:\left({n}\geqslant\mathrm{1}\right) \\ $$

Question Number 116529 Answers: 1 Comments: 1

Question Number 116524 Answers: 1 Comments: 1

if α+β+γ=180° prove: sin α+sin β+sin γ+sin (π/3) < 4 sin (π/3)

$$\mathrm{if}\:\alpha+\beta+\gamma=\mathrm{180}° \\ $$$$\mathrm{prove}: \\ $$$$\mathrm{sin}\:\alpha+\mathrm{sin}\:\beta+\mathrm{sin}\:\gamma+\mathrm{sin}\:\frac{\pi}{\mathrm{3}}\:<\:\mathrm{4}\:\mathrm{sin}\:\frac{\pi}{\mathrm{3}} \\ $$

Question Number 116523 Answers: 2 Comments: 0

Question Number 116520 Answers: 1 Comments: 0

find the center of circle with radius R and tangent by sides AB and BC of triangle?

$${find}\:{the}\:{center}\:{of}\:{circle}\:{with}\:{radius}\:{R} \\ $$$${and}\:{tangent}\:{by}\:{sides}\:{AB}\:{and}\:{BC}\:{of}\: \\ $$$${triangle}? \\ $$

Question Number 116516 Answers: 2 Comments: 0

Question Number 116513 Answers: 1 Comments: 0

give a∈Z and prove that if (a−2)∣4 than (a−3)can′t∣6

$${give}\:{a}\in{Z}\:{and}\:{prove}\:{that}\:{if}\:\left({a}−\mathrm{2}\right)\mid\mathrm{4}\:{than}\:\left({a}−\mathrm{3}\right){can}'{t}\mid\mathrm{6} \\ $$

Question Number 116512 Answers: 0 Comments: 1

give x∈R and prove that if x=(√(2x+3)) than x=3

$${give}\:{x}\in{R}\:{and}\:{prove}\:{that}\:{if}\:\:{x}=\sqrt{\mathrm{2}{x}+\mathrm{3}}\:{than}\:{x}=\mathrm{3} \\ $$

Question Number 116509 Answers: 2 Comments: 0

x^4 −48x^2 +x+565=0 x=?

$$\mathrm{x}^{\mathrm{4}} −\mathrm{48x}^{\mathrm{2}} +\mathrm{x}+\mathrm{565}=\mathrm{0}\: \\ $$$$\mathrm{x}=? \\ $$

Question Number 116507 Answers: 1 Comments: 0

Question Number 116503 Answers: 2 Comments: 0

If tan α and tan β are the roots of equation 2x^2 −5x+1=0 find { ((tan ((α/2)+2β))),((tan (2α−(β/2)))) :}

$$\mathrm{If}\:\mathrm{tan}\:\alpha\:\mathrm{and}\:\mathrm{tan}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\: \\ $$$$\mathrm{of}\:\mathrm{equation}\:\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{find}\:\begin{cases}{\mathrm{tan}\:\left(\frac{\alpha}{\mathrm{2}}+\mathrm{2}\beta\right)}\\{\mathrm{tan}\:\left(\mathrm{2}\alpha−\frac{\beta}{\mathrm{2}}\right)}\end{cases} \\ $$

Question Number 116501 Answers: 1 Comments: 0

∫ ((sin x−cos x)/( (√(sin 2x)))) dx ?

$$\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx}\:? \\ $$

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