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Question Number 90575 Answers: 0 Comments: 0

Solve the differential equation: x y_3 +(1−2x^2 )y_2 −8x _ y_1 −4y= e^x

$$\:\boldsymbol{\mathrm{Solve}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{differential}}\:\boldsymbol{\mathrm{equation}}: \\ $$$$\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{y}}_{\mathrm{3}} +\left(\mathrm{1}−\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\boldsymbol{\mathrm{y}}_{\mathrm{2}} −\mathrm{8}\boldsymbol{\mathrm{x}}\:_{\:} \boldsymbol{\mathrm{y}}_{\mathrm{1}} −\mathrm{4}\boldsymbol{\mathrm{y}}=\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} \\ $$

Question Number 90574 Answers: 0 Comments: 2

Use gamma function to prove (i) . ∫_0 ^( (𝛑/8)) cos^3 4x dx= (1/6). (ii). ∫_0 ^( (𝛑/6)) cos^4 3𝛉 sin^2 6𝛉 d𝛉 = ((5𝛑)/(192)).

Question Number 90570 Answers: 1 Comments: 1

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

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

Question Number 90566 Answers: 0 Comments: 0

∫_0 ^(infinity) Sin^4 3x/x^2 dx

$$\int_{\mathrm{0}} ^{\mathrm{infinity}} \mathrm{Sin}^{\mathrm{4}} \mathrm{3x}/\mathrm{x}^{\mathrm{2}} \mathrm{dx} \\ $$

Question Number 90564 Answers: 0 Comments: 1

find lim_(x→0) (((^3 (√(1+cos(2x)))−(^3 (√2)))/(x^2 sin(3x)))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\left(^{\mathrm{3}} \sqrt{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)}−\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)\right.}{{x}^{\mathrm{2}} {sin}\left(\mathrm{3}{x}\right)} \\ $$

Question Number 90562 Answers: 0 Comments: 2

prove that Σ_(n=1) ^∞ (1/(2n(2n−1)))=ln2

$${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}{n}\left(\mathrm{2}{n}−\mathrm{1}\right)}={ln}\mathrm{2} \\ $$

Question Number 90561 Answers: 0 Comments: 2

prove that Π_(n=2) ^∞ (1−(1/n^2 ))=(1/2)

$${prove}\:{that} \\ $$$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 90557 Answers: 0 Comments: 7

Find the area enclose by the line y = x − 1 and the parabola y^2 = 2x + 6

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{enclose}\:\mathrm{by}\:\mathrm{the}\:\mathrm{line}\:\:\:\mathrm{y}\:\:=\:\:\mathrm{x}\:\:−\:\:\mathrm{1}\:\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{parabola}\:\:\:\mathrm{y}^{\mathrm{2}} \:\:=\:\:\mathrm{2x}\:\:+\:\:\mathrm{6} \\ $$

Question Number 90555 Answers: 1 Comments: 2

Question Number 90581 Answers: 1 Comments: 0

given that α and β are roots of the equation aχ^2 +bχ+c=0. show that λμb^2 =ac(λ+μ)^(2 ) where (α/β)=(λ/μ)

$${given}\:{that}\:\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\:\:{the}\:{equation}\: \\ $$$${a}\chi^{\mathrm{2}} +{b}\chi+{c}=\mathrm{0}.\:{show}\:{that}\:\lambda\mu{b}^{\mathrm{2}} ={ac}\left(\lambda+\mu\right)^{\mathrm{2}\:} \\ $$$${where}\:\frac{\alpha}{\beta}=\frac{\lambda}{\mu} \\ $$

Question Number 90550 Answers: 0 Comments: 3

x^4 + (1/x^4 ) = 527 (x−1)(x−2)(x−3)(x−4) ?

$${x}^{\mathrm{4}} \:+\:\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:=\:\mathrm{527}\: \\ $$$$\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)\:?\: \\ $$

Question Number 90544 Answers: 1 Comments: 0

∫ (dx/(√(2−cos x)))

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

Question Number 90535 Answers: 1 Comments: 0

In a triangle ABC, a(b cos C−c cos B)=

$$\mathrm{In}\:\mathrm{a}\:\mathrm{triangle}\:{ABC},\:{a}\left({b}\:\mathrm{cos}\:{C}−{c}\:\mathrm{cos}\:{B}\right)= \\ $$

Question Number 90531 Answers: 0 Comments: 2

Please help me to find the value of ′n′ C_8 ^(n+3) =C_(20) ^(n+3)

$${Please}\:{help}\:{me}\:{to}\:{find}\:{the}\:{value} \\ $$$${of}\:\:\:'{n}' \\ $$$${C}_{\mathrm{8}} ^{{n}+\mathrm{3}} ={C}_{\mathrm{20}} ^{{n}+\mathrm{3}} \\ $$

Question Number 90520 Answers: 2 Comments: 1

Question Number 90514 Answers: 2 Comments: 6

Question Number 90512 Answers: 1 Comments: 2

∫_0 ^1 (1/x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}}{dx} \\ $$

Question Number 90509 Answers: 2 Comments: 3

Question Number 90508 Answers: 0 Comments: 0

Σ_(n=1) ^∞ ((1/(n^m (1+n)^m ))) what is the general for this sum

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}^{{m}} \left(\mathrm{1}+{n}\right)^{{m}} }\right) \\ $$$${what}\:{is}\:{the}\:{general}\:{for}\:{this}\:{sum} \\ $$

Question Number 90507 Answers: 0 Comments: 2

if 2^(sin x) +2^(cos x) =2^(1+(1/(√2))) then find the value of x=?

$$ \\ $$$$\:\mathrm{if}\:\:\mathrm{2}^{\mathrm{sin}\:\mathrm{x}} +\mathrm{2}^{\mathrm{cos}\:\mathrm{x}} =\mathrm{2}^{\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}} \:\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}=? \\ $$

Question Number 90503 Answers: 0 Comments: 2

Question Number 90489 Answers: 0 Comments: 0

Find f(x) if it equals Σ_(n=0) ^∞ (n^x /(n!))

$${Find}\:{f}\left({x}\right)\:{if}\:{it}\:{equals}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{{x}} }{{n}!} \\ $$

Question Number 90485 Answers: 0 Comments: 1

if x_0 =x_1 =1 and x_(n+1) =1996x_n +1997x_(n−1) for n≥2. Find the remainder of the division of x_(1996) by 3

$${if}\:{x}_{\mathrm{0}} ={x}_{\mathrm{1}} =\mathrm{1}\:{and}\:{x}_{{n}+\mathrm{1}} =\mathrm{1996}{x}_{{n}} +\mathrm{1997}{x}_{{n}−\mathrm{1}} \\ $$$${for}\:{n}\geqslant\mathrm{2}.\:{Find}\:{the}\:{remainder}\:{of} \\ $$$${the}\:{division}\:{of}\:{x}_{\mathrm{1996}} \:{by}\:\mathrm{3} \\ $$

Question Number 90483 Answers: 0 Comments: 0

prove that/ ((sin^3 a)/(sin b))+((cos^3 a)/(cos b))≥sec(a−b) for all a,b∈ (0,(π/2))

$${prove}\:{that}/\:\frac{{sin}^{\mathrm{3}} {a}}{{sin}\:{b}}+\frac{{cos}^{\mathrm{3}} {a}}{{cos}\:{b}}\geqslant{sec}\left({a}−{b}\right) \\ $$$${for}\:{all}\:{a},{b}\in\:\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right) \\ $$

Question Number 90480 Answers: 0 Comments: 0

∫e^x (((1+cos(x))(1−sin(x)))/((e^x cos(x)+1)^2 ))dx

$$\int{e}^{{x}} \frac{\left(\mathrm{1}+{cos}\left({x}\right)\right)\left(\mathrm{1}−{sin}\left({x}\right)\right)}{\left({e}^{{x}} \:{cos}\left({x}\right)+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 90475 Answers: 0 Comments: 3

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