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Question Number 37382    Answers: 2   Comments: 3

find all real solutions (2−x^2 )^(x^2 −3(√(2x))+4) =1 i}what if x is permitted to be complex number ii}what if 1=(−1)^(2n) ?

$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{real}}\:\boldsymbol{\mathrm{solutions}} \\ $$$$\left(\mathrm{2}−\boldsymbol{{x}}^{\mathrm{2}} \right)^{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{3}\sqrt{\mathrm{2}\boldsymbol{{x}}}+\mathrm{4}} =\mathrm{1} \\ $$$$\left.\mathrm{i}\right\}\boldsymbol{\mathrm{what}}\:\boldsymbol{\mathrm{if}}\:\boldsymbol{{x}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{permitted}} \\ $$$$\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{complex}}\:\boldsymbol{\mathrm{number}} \\ $$$$\left.\boldsymbol{\mathrm{ii}}\right\}\boldsymbol{\mathrm{what}}\:\boldsymbol{\mathrm{if}}\:\mathrm{1}=\left(−\mathrm{1}\right)^{\mathrm{2}\boldsymbol{\mathrm{n}}} ? \\ $$

Question Number 37367    Answers: 0   Comments: 1

Question Number 37366    Answers: 0   Comments: 0

let f(x) = (e^(−(x/a)) /a) find L(f(x)).

$${let}\:{f}\left({x}\right)\:=\:\frac{{e}^{−\frac{{x}}{{a}}} }{{a}} \\ $$$${find}\:{L}\left({f}\left({x}\right)\right). \\ $$

Question Number 37365    Answers: 0   Comments: 1

find L^(−1) { (1/((a+x)^2 ))} and L^(−1) {(1/((a+x)^3 ))} .

$${find}\:{L}^{−\mathrm{1}} \left\{\:\:\frac{\mathrm{1}}{\left({a}+{x}\right)^{\mathrm{2}} }\right\}\:\:{and}\:{L}^{−\mathrm{1}} \left\{\frac{\mathrm{1}}{\left({a}+{x}\right)^{\mathrm{3}} }\right\}\:. \\ $$

Question Number 37364    Answers: 0   Comments: 1

calculate L{ ((x^(n−1) e^(−ax) )/((n−1)!))} then conclude L^(−1) { (1/((a+x)^n ))}

$${calculate}\:\:{L}\left\{\:\frac{{x}^{{n}−\mathrm{1}} \:{e}^{−{ax}} }{\left({n}−\mathrm{1}\right)!}\right\}\:{then}\:{conclude} \\ $$$${L}^{−\mathrm{1}} \left\{\:\:\frac{\mathrm{1}}{\left({a}+{x}\right)^{{n}} }\right\} \\ $$

Question Number 37363    Answers: 1   Comments: 0

solve y^′ +xe^(−x^2 ) y =e^(−x) .

$${solve}\:{y}^{'} \:\:+{xe}^{−{x}^{\mathrm{2}} } {y}\:\:={e}^{−{x}} \:\:. \\ $$

Question Number 37362    Answers: 0   Comments: 1

find L(cos(wx)) and L(sin(wx)) L is laplace transform .

$${find}\:{L}\left({cos}\left({wx}\right)\right)\:{and}\:{L}\left({sin}\left({wx}\right)\right) \\ $$$${L}\:{is}\:{laplace}\:{transform}\:\:. \\ $$

Question Number 37361    Answers: 0   Comments: 0

calculate ∫_0 ^(+∞) ((ln(x))/(1+x^3 )) .

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

Question Number 37360    Answers: 0   Comments: 1

calculate ∫_0 ^(+∞) (dx/(1+x^3 ))

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} } \\ $$

Question Number 37359    Answers: 0   Comments: 1

let g(x)= ((ln(z))/(1+z^3 )) give the poles z_i of g and calculate Res(g ,z_i )

$${let}\:\:\:{g}\left({x}\right)=\:\frac{{ln}\left({z}\right)}{\mathrm{1}+{z}^{\mathrm{3}} }\:\:{give}\:{the}\:{poles}\:{z}_{{i}} \:{of}\:{g}\:{and} \\ $$$${calculate}\:{Res}\left({g}\:,{z}_{{i}} \right)\: \\ $$$$ \\ $$

Question Number 37357    Answers: 0   Comments: 2

let a>0 b from C and Re(b)>0 1) calculate ∫_(−∞) ^(+∞) ((b cos(ax))/(x^2 +b^2 ))dx 2) find the value of ∫_(−∞) ^(+∞) ((x sin(ax))/(x^2 +b^2 )) dx.

$${let}\:{a}>\mathrm{0}\:{b}\:{from}\:{C}\:{and}\:\:{Re}\left({b}\right)>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{b}\:{cos}\left({ax}\right)}{{x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}\:{sin}\left({ax}\right)}{{x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} }\:{dx}. \\ $$

Question Number 37356    Answers: 0   Comments: 2

let b ∈C and Re(b) >0 prove that ∫_(−∞) ^(+∞) (e^(iax) /(x−ib))dx =2iπ e^(−ab ) and ∫_(−∞) ^(+∞) (e^(iax) /(x+ib)) dx =0

$${let}\:\:{b}\:\in{C}\:\:{and}\:{Re}\left({b}\right)\:>\mathrm{0}\:{prove}\:{that} \\ $$$$\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{iax}} }{{x}−{ib}}{dx}\:=\mathrm{2}{i}\pi\:{e}^{−{ab}\:\:} \:\:\:{and} \\ $$$$\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{iax}} }{{x}+{ib}}\:{dx}\:=\mathrm{0} \\ $$

Question Number 37355    Answers: 0   Comments: 0

let Σ a_n x^n with radius of convergence R prove that R = (1/(lim_(n→+∞) sup^n (√(∣a_n ∣)))) .

$$\:{let}\:\Sigma\:{a}_{{n}} {x}^{{n}} \:\:\:{with}\:{radius}\:{of}\:{convergence}\:{R} \\ $$$${prove}\:{that}\:{R}\:=\:\frac{\mathrm{1}}{{lim}_{{n}\rightarrow+\infty} \:{sup}^{{n}} \sqrt{\mid{a}_{{n}} \mid}}\:\:. \\ $$

Question Number 37354    Answers: 0   Comments: 1

let f(z) = ((z^2 +1)/((z^2 −1)(z^2 −4))) developp f at integr serie .

$${let}\:{f}\left({z}\right)\:=\:\:\frac{{z}^{\mathrm{2}} \:+\mathrm{1}}{\left({z}^{\mathrm{2}} \:−\mathrm{1}\right)\left({z}^{\mathrm{2}} \:−\mathrm{4}\right)} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 37353    Answers: 0   Comments: 1

let g(z) =(z/(e^z −1)) developp g at integr serie .

$${let}\:{g}\left({z}\right)\:=\frac{{z}}{{e}^{{z}} −\mathrm{1}} \\ $$$${developp}\:{g}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 37352    Answers: 0   Comments: 1

let f(z) = e^(−(1/z^2 )) 1) give f(z) at form of serie 2) give ∫_1 ^2 f(z)dz at form of serie .

$${let}\:\:{f}\left({z}\right)\:=\:{e}^{−\frac{\mathrm{1}}{{z}^{\mathrm{2}} }} \:\: \\ $$$$\left.\mathrm{1}\right)\:{give}\:{f}\left({z}\right)\:{at}\:{form}\:{of}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{give}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} {f}\left({z}\right){dz}\:\:\:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 37350    Answers: 0   Comments: 2

fond the value of ∫_0 ^(2π) (dt/((a +cost)^2 )) with a>1.

$${fond}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{\left({a}\:+{cost}\right)^{\mathrm{2}} }\:\:{with}\:{a}>\mathrm{1}. \\ $$$$ \\ $$

Question Number 37349    Answers: 0   Comments: 1

calculate ∫_0 ^(2π) (dt/(1−2pcost +p^2 )) if ∣p∣<1

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{{dt}}{\mathrm{1}−\mathrm{2}{pcost}\:+{p}^{\mathrm{2}} }\:\:{if}\:\mid{p}\mid<\mathrm{1} \\ $$

Question Number 37348    Answers: 0   Comments: 2

calculate ∫_0 ^(2π) (dt/(p +cost)) with p>1

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{dt}}{{p}\:+{cost}}\:\:{with}\:{p}>\mathrm{1} \\ $$

Question Number 37347    Answers: 1   Comments: 2

let r =(√(p^2 +q^2 )) p and q from R and p>0 q>0 1)prove that ∫_0 ^(+∞) e^(−px) ((cos(px))/(√x))dx=((√π)/r)(√((r+p)/2)) 2) ∫_0 ^∞ e^(−px) ((sin(qx))/(√x))dx =((√π)/r) (√((r−p)/2))

$${let}\:{r}\:=\sqrt{{p}^{\mathrm{2}} \:+{q}^{\mathrm{2}} }\:\:\:{p}\:{and}\:{q}\:{from}\:{R}\:\:{and}\:{p}>\mathrm{0}\:\:{q}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:{e}^{−{px}} \:\frac{{cos}\left({px}\right)}{\sqrt{{x}}}{dx}=\frac{\sqrt{\pi}}{{r}}\sqrt{\frac{{r}+{p}}{\mathrm{2}}} \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{px}} \:\:\frac{{sin}\left({qx}\right)}{\sqrt{{x}}}{dx}\:=\frac{\sqrt{\pi}}{{r}}\:\sqrt{\frac{{r}−{p}}{\mathrm{2}}} \\ $$

Question Number 37346    Answers: 0   Comments: 0

find the value of ∫_0 ^(2π) (dt/(a cos^2 t +b sin^2 t)) with a>0 and b>0 .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{a}\:{cos}^{\mathrm{2}} {t}\:+{b}\:{sin}^{\mathrm{2}} {t}} \\ $$$${with}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:. \\ $$

Question Number 37345    Answers: 0   Comments: 0

calculate I(a) = ∫_0 ^(2π) ((1+acost)/(1+2acost +a^2 ))dt 1) if ∣a∣<1 2) if ∣a∣>1

$${calculate}\:{I}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{\mathrm{1}+{acost}}{\mathrm{1}+\mathrm{2}{acost}\:+{a}^{\mathrm{2}} }{dt}\:\: \\ $$$$\left.\mathrm{1}\right)\:{if}\:\:\mid{a}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{if}\:\mid{a}\mid>\mathrm{1} \\ $$

Question Number 37344    Answers: 0   Comments: 1

solve the d.e. y^′ −xy =cosx .

$${solve}\:{the}\:{d}.{e}.\:{y}^{'} \:−{xy}\:\:={cosx}\:. \\ $$

Question Number 37343    Answers: 0   Comments: 3

let f(x) = ∫_0 ^1 ln(1+xt^2 )dt with ∣x∣<1 1) find f(x) 2) calculate ∫_0 ^1 ln(2+t^2 )dt .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right){dt}\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{2}+{t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 37342    Answers: 0   Comments: 2

calculate Σ_(n=1) ^∞ (((−1)^n )/(n^2 (n+1))) x^n with ∣x∣<1 2) find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)2^n )) .

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:{x}^{{n}} \:\:\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)\mathrm{2}^{{n}} }\:. \\ $$

Question Number 37341    Answers: 0   Comments: 1

calculate Σ_(n=1) ^∞ (3/(n^2 (2n+1)^2 ))

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

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