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Question Number 37462 Answers: 1 Comments: 0

If x+3 is the common factor of the expressions ax^2 +bx+1 and px^2 +qx−3, then ((−(9a+3p))/(3b+q)) = ____.

$$\mathrm{If}\:{x}+\mathrm{3}\:\mathrm{is}\:\mathrm{the}\:\mathrm{common}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{expressions}\:{ax}^{\mathrm{2}} +{bx}+\mathrm{1}\:\mathrm{and}\: \\ $$$${px}^{\mathrm{2}} +{qx}−\mathrm{3},\:\mathrm{then}\:\frac{−\left(\mathrm{9}{a}+\mathrm{3}{p}\right)}{\mathrm{3}{b}+{q}}\:=\:\_\_\_\_. \\ $$

Question Number 37451 Answers: 2 Comments: 1

∫_0 ^( 2π) ((a^2 sin^2 θdθ)/(√(a^2 sin^2 θ+b^2 cos^2 θ))) = ?

$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \frac{{a}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta{d}\theta}{\sqrt{{a}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta+{b}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \theta}}\:=\:? \\ $$

Question Number 37449 Answers: 1 Comments: 0

∫_0 ^1 ((3x^3 −x^2 +2x+4)/(√(x^2 −3x+2))) dx=

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{4}}{\sqrt{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}}\:{dx}= \\ $$

Question Number 37447 Answers: 2 Comments: 5

Question Number 37432 Answers: 2 Comments: 0

∫(dα/((1/2)tan α +2cot (α/2)))=? ∫(dβ/(2tan (β/2) +(1/2)cot β))=?

$$\int\frac{{d}\alpha}{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{tan}\:\alpha\:+\mathrm{2cot}\:\frac{\alpha}{\mathrm{2}}}=? \\ $$$$\int\frac{{d}\beta}{\mathrm{2tan}\:\frac{\beta}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cot}\:\beta}=? \\ $$

Question Number 37425 Answers: 1 Comments: 0

∫((x^5 −3x^4 −23x^3 +51x^2 +94x−120)/(8x^3 (√(42+x−x^2 ))))dx=?

$$\int\frac{{x}^{\mathrm{5}} −\mathrm{3}{x}^{\mathrm{4}} −\mathrm{23}{x}^{\mathrm{3}} +\mathrm{51}{x}^{\mathrm{2}} +\mathrm{94}{x}−\mathrm{120}}{\mathrm{8}{x}^{\mathrm{3}} \sqrt{\mathrm{42}+{x}−{x}^{\mathrm{2}} }}{dx}=? \\ $$

Question Number 37424 Answers: 0 Comments: 0

solve for Z(x,t)if Z_n =x^2 t subjected Z(x,0)=x^2 and Z(1,t)=cost

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{Z}}\left(\boldsymbol{{x}},\boldsymbol{{t}}\right)\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{Z}}_{\boldsymbol{\mathrm{n}}} =\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{t}}\:\:\boldsymbol{\mathrm{subjected}}\:\boldsymbol{{Z}}\left(\boldsymbol{{x}},\mathrm{0}\right)=\boldsymbol{{x}}^{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{{Z}}\left(\mathrm{1},\boldsymbol{{t}}\right)=\boldsymbol{\mathrm{cos}{t}} \\ $$

Question Number 37422 Answers: 0 Comments: 0

show that U(x,y)=F(2x+5y)+G(x−5y) is the general solution 4U_υ −25U_υ =0

$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{{U}}\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)=\boldsymbol{{F}}\left(\mathrm{2}\boldsymbol{{x}}+\mathrm{5}\boldsymbol{{y}}\right)+\boldsymbol{{G}}\left(\boldsymbol{{x}}−\mathrm{5}\boldsymbol{{y}}\right) \\ $$$$\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{general}}\:\boldsymbol{\mathrm{solution}}\:\mathrm{4}\boldsymbol{{U}}_{\upsilon} −\mathrm{25}\boldsymbol{{U}}_{\upsilon} =\mathrm{0} \\ $$

Question Number 37630 Answers: 1 Comments: 0

find ∫ (dx/((√x) +(√(x+1)) +(√(x+2))))

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

Question Number 37414 Answers: 1 Comments: 0

x^x =2 find the value of x.

$${x}^{{x}} =\mathrm{2} \\ $$$$ \\ $$$${find}\:{the}\:{value}\:{of}\:{x}. \\ $$

Question Number 37411 Answers: 1 Comments: 0

((√(2)^(√2) ))

$$\left(\sqrt{\left.\mathrm{2}\right)^{\sqrt{\mathrm{2}}} }\right. \\ $$

Question Number 37404 Answers: 1 Comments: 0

∫_0 ^( 2π) (√(a^2 +b^2 −2abcos θ)) dθ with a>b>0 .

$$\int_{\mathrm{0}} ^{\:\:\mathrm{2}\pi} \sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −\mathrm{2}{ab}\mathrm{cos}\:\theta}\:{d}\theta\: \\ $$$$\:\:\:\:{with}\:\:{a}>{b}>\mathrm{0}\:. \\ $$

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) ?

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 .

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

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