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Question Number 143262 Answers: 1 Comments: 0

developp at fourier serie f(x)=(1/(cosx +2sinx))

$${developp}\:{at}\:{fourier}\:{serie} \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{{cosx}\:+\mathrm{2}{sinx}} \\ $$

Question Number 143261 Answers: 1 Comments: 0

find Y_n =∫_0 ^∞ (dx/((x+1)(x+2)....(x+n))) (n>1 integr)

$${find}\:{Y}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)....\left({x}+{n}\right)} \\ $$$$\left({n}>\mathrm{1}\:{integr}\right) \\ $$

Question Number 143260 Answers: 0 Comments: 1

find ∫ (dx/( (√x)+(√(x+1))+(√(x+2))))

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

Question Number 143259 Answers: 1 Comments: 0

solve y^(′′) −y^′ +2=xsin(3x)

$${solve}\:{y}^{''} −{y}^{'} +\mathrm{2}={xsin}\left(\mathrm{3}{x}\right) \\ $$

Question Number 143258 Answers: 1 Comments: 0

calculate lim_(x→1) ∫_x ^x^2 ((sh(xt))/(x+t))dt

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{1}} \int_{{x}} ^{{x}^{\mathrm{2}} } \:\frac{{sh}\left({xt}\right)}{{x}+{t}}{dt} \\ $$

Question Number 143257 Answers: 0 Comments: 0

find ∫∫_([0,1]) e^(−(x^2 +y^2 )) arctan(2(√(x^2 +y^2 )))dxdy

$${find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]} {e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} {arctan}\left(\mathrm{2}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right){dxdy} \\ $$

Question Number 143256 Answers: 0 Comments: 0

calculate ∫_0 ^1 ∫_x ^(2−x) e^(−xy) (√(x+y))dy dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \int_{{x}} ^{\mathrm{2}−{x}} {e}^{−{xy}} \sqrt{{x}+{y}}{dy}\:{dx} \\ $$

Question Number 143255 Answers: 2 Comments: 0

find lim_(x→0) ((sin(1−cosx)+1−cos(sinx))/x^2 )

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

Question Number 143254 Answers: 1 Comments: 0

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

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

Question Number 143253 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(x^3 ))/(1+x^3 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({x}^{\mathrm{3}} \right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 143252 Answers: 1 Comments: 0

lim_(x→0) ((8x−6x sin x+sin 2x)/x^5 ) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{8}{x}−\mathrm{6}{x}\:\mathrm{sin}\:{x}+\mathrm{sin}\:\mathrm{2}{x}}{{x}^{\mathrm{5}} }\:=? \\ $$

Question Number 143251 Answers: 4 Comments: 0

Question Number 143248 Answers: 1 Comments: 1

∫(dx/(x^2 −4x+1))

$$\int\frac{{dx}}{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{1}} \\ $$

Question Number 143240 Answers: 1 Comments: 0

(2/y)=(1/x)((a/y)+1) x=2(4a−y) Is equations system

$$\frac{\mathrm{2}}{{y}}=\frac{\mathrm{1}}{{x}}\left(\frac{{a}}{{y}}+\mathrm{1}\right) \\ $$$${x}=\mathrm{2}\left(\mathrm{4}{a}−{y}\right) \\ $$$${Is}\:{equations}\:{system} \\ $$

Question Number 143239 Answers: 0 Comments: 0

Question Number 143238 Answers: 1 Comments: 0

Question Number 143234 Answers: 1 Comments: 0

Question Number 143231 Answers: 0 Comments: 0

.....mathematical ......Analysis.... if :: 𝛗(n):=∫_0 ^( 1) x^(2n−1) log(1+x)dx then find the value of :: Θ:= Σ_(n=1) ^∞ (−1)^n 𝛗(n) .......m.n

$$.....{mathematical}\:......{Analysis}.... \\ $$$$\:{if}\:::\:\:\:\boldsymbol{\phi}\left({n}\right):=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}{n}−\mathrm{1}} {log}\left(\mathrm{1}+{x}\right){dx} \\ $$$$\:{then}\:\:{find}\:\:{the}\:{value}\:{of}\:\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Theta:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \boldsymbol{\phi}\left({n}\right)\: \\ $$$$\:\:\:\:\:\:.......{m}.{n} \\ $$

Question Number 143230 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (x^(3n+1) /(3n+1))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{x}^{\mathrm{3n}+\mathrm{1}} }{\mathrm{3n}+\mathrm{1}}=? \\ $$

Question Number 143229 Answers: 0 Comments: 0

Let a,b,c ≥ 0 and (1+a)(1+b)(1+c) = 8. Prove that (a+((2b+1)/(a+b+1)))(b+((2c+1)/(b+c+1)))(c+((2a+1)/(c+a+1))) ≥ 8

$$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{b}\right)\left(\mathrm{1}+{c}\right)\:=\:\mathrm{8}. \\ $$$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+\frac{\mathrm{2}{b}+\mathrm{1}}{{a}+{b}+\mathrm{1}}\right)\left({b}+\frac{\mathrm{2}{c}+\mathrm{1}}{{b}+{c}+\mathrm{1}}\right)\left({c}+\frac{\mathrm{2}{a}+\mathrm{1}}{{c}+{a}+\mathrm{1}}\right)\:\geqslant\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 143225 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (1/((2021^n )(n!))) = Σ_(n=0) ^∞ (1/((2021^n )(∫_0 ^( ∞) t^n .e^(−t) dt)))

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\left(\mathrm{2021}^{{n}} \right)\left({n}!\right)}\:=\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\:\frac{\mathrm{1}}{\left(\mathrm{2021}^{{n}} \right)\left(\int_{\mathrm{0}} ^{\:\infty} {t}^{{n}} .{e}^{−{t}} \:\:{dt}\right)} \\ $$

Question Number 143222 Answers: 1 Comments: 0

If z=cos θ+i sin θ, by expand (z+(1/z))^4 (z−(1/z))^4 or other method, prove 128 sin^4 θcos^4 θ=cos 8θ−4cos 4θ+3.

$$\mathrm{If}\:{z}=\mathrm{cos}\:\theta+{i}\:\mathrm{sin}\:\theta,\:\mathrm{by}\:\mathrm{expand} \\ $$$$\left({z}+\frac{\mathrm{1}}{{z}}\right)^{\mathrm{4}} \left({z}−\frac{\mathrm{1}}{{z}}\right)^{\mathrm{4}} \mathrm{or}\:\mathrm{other}\:\mathrm{method}, \\ $$$$\mathrm{prove}\:\mathrm{128}\:\mathrm{sin}^{\mathrm{4}} \theta\mathrm{cos}^{\mathrm{4}} \theta=\mathrm{cos}\:\mathrm{8}\theta−\mathrm{4cos}\:\mathrm{4}\theta+\mathrm{3}. \\ $$

Question Number 143219 Answers: 0 Comments: 1

Question Number 143210 Answers: 1 Comments: 0

∫_0 ^1 ((7^(x+1) +3^(x+1) )/(x+1))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{7}^{{x}+\mathrm{1}} +\mathrm{3}^{{x}+\mathrm{1}} }{{x}+\mathrm{1}}{dx} \\ $$

Question Number 143208 Answers: 1 Comments: 0

3^x +4x−3=x^4 find x

$$\mathrm{3}^{{x}} +\mathrm{4}{x}−\mathrm{3}={x}^{\mathrm{4}} \\ $$$${find}\:{x} \\ $$

Question Number 143200 Answers: 1 Comments: 0

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