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AllQuestion and Answers: Page 1500

Question Number 60697 Answers: 1 Comments: 1

Question Number 60695 Answers: 0 Comments: 1

solve (√(3+x^2 ))y^(′′) −(2x+1)y^′ =x^2 e^(−x^2 )

$${solve}\:\sqrt{\mathrm{3}+{x}^{\mathrm{2}} }{y}^{''} \:\:\:\:−\left(\mathrm{2}{x}+\mathrm{1}\right){y}^{'} \:={x}^{\mathrm{2}} \:{e}^{−{x}^{\mathrm{2}} \:\:\:} \\ $$

Question Number 60694 Answers: 0 Comments: 0

solve (x^3 −x)y^(′′) −2x y^′ +3y =xln(1+x)

$${solve}\:\:\left({x}^{\mathrm{3}} −{x}\right){y}^{''} \:\:\:−\mathrm{2}{x}\:{y}^{'} \:+\mathrm{3}{y}\:={xln}\left(\mathrm{1}+{x}\right) \\ $$

Question Number 60693 Answers: 0 Comments: 0

solve (2+e^(−x) )y^′ +(2x+e^x )y =e^x sinx

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

Question Number 60692 Answers: 0 Comments: 0

find ∫ arctan(2cosx)dx

$${find}\:\:\int\:\:{arctan}\left(\mathrm{2}{cosx}\right){dx}\: \\ $$

Question Number 60691 Answers: 0 Comments: 1

calculate f(a) = ∫ (1−(a/x^2 )) arctan(x+(a/x))dx with a real .

$${calculate}\:{f}\left({a}\right)\:=\:\int\:\:\:\left(\mathrm{1}−\frac{{a}}{{x}^{\mathrm{2}} }\right)\:{arctan}\left({x}+\frac{{a}}{{x}}\right){dx}\:\:\:{with}\:{a}\:{real}\:. \\ $$

Question Number 60690 Answers: 0 Comments: 0

find ∫ (x+3)(√((1−x^2 )/(1+x^2 )))dx

$${find}\:\int\:\left({x}+\mathrm{3}\right)\sqrt{\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }}{dx}\: \\ $$

Question Number 60688 Answers: 0 Comments: 0

find ∫ e^(−x) (√((3−x)/(3+x)))dx

$${find}\:\int\:\:\:{e}^{−{x}} \sqrt{\frac{\mathrm{3}−{x}}{\mathrm{3}+{x}}}{dx} \\ $$

Question Number 60687 Answers: 1 Comments: 2

calculate ∫ (((√(1+x^2 ))−2x)/((√(1+x^2 )) +2x)) dx

$${calculate}\:\:\int\:\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }−\mathrm{2}{x}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\mathrm{2}{x}}\:{dx} \\ $$

Question Number 60685 Answers: 1 Comments: 1

find I_n =∫_0 ^(π/2) ((1−cos(nx))/(sin^2 (nx)))dx

$${find}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{1}−{cos}\left({nx}\right)}{{sin}^{\mathrm{2}} \left({nx}\right)}{dx}\: \\ $$

Question Number 60686 Answers: 0 Comments: 0

let f(x) =cos(2x) ,2π periodic , developp f at fourier serie

$${let}\:{f}\left({x}\right)\:={cos}\left(\mathrm{2}{x}\right)\:\:\:\:,\mathrm{2}\pi\:{periodic}\:,\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$

Question Number 60683 Answers: 0 Comments: 0

find A_n =∫_0 ^(π/4) sin^n xdx with n integr natural .

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{sin}^{{n}} {xdx}\:\:\:\:{with}\:{n}\:{integr}\:{natural}\:. \\ $$

Question Number 60682 Answers: 0 Comments: 1

simplify S_n =Σ_(k=0) ^n sin^k (x)cos(kx)

$${simplify}\:\:\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\:{sin}^{{k}} \left({x}\right){cos}\left({kx}\right)\:\:\: \\ $$

Question Number 60681 Answers: 0 Comments: 1

calculate L(e^(−2x) sin(αx)) α real and L laplace transform

$${calculate}\:\:{L}\left({e}^{−\mathrm{2}{x}} {sin}\left(\alpha{x}\right)\right)\:\:\:\:\alpha\:{real}\:\:\:{and}\:{L}\:{laplace}\:{transform} \\ $$

Question Number 60680 Answers: 0 Comments: 2

study the integral ∫_0 ^1 (x/(ln(1−x)))dx

$${study}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}}{{ln}\left(\mathrm{1}−{x}\right)}{dx} \\ $$

Question Number 60679 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((ln(1+e^(−x^2 ) ))/(x^2 +4)) dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{e}^{−{x}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}\:{dx} \\ $$

Question Number 60678 Answers: 0 Comments: 3

calculate ∫_0 ^1 ((ln(1−x^2 ))/x) dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}}\:{dx} \\ $$

Question Number 60677 Answers: 0 Comments: 0

let S_n =Σ_(k=1) ^n sin(((k^2 π)/n^3 )) determine lim_(n→∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{{k}^{\mathrm{2}} \pi}{{n}^{\mathrm{3}} }\right)\:\:{determine}\:{lim}_{{n}\rightarrow\infty} \:\:{S}_{{n}} \\ $$

Question Number 60676 Answers: 1 Comments: 2

let S_n =Σ_(k=1) ^n sin^2 (((kπ)/n^2 )) find lim_(n→∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:{sin}^{\mathrm{2}} \left(\frac{{k}\pi}{{n}^{\mathrm{2}} }\right)\:\:\:\:\:{find}\:{lim}_{{n}\rightarrow\infty} \:\:{S}_{{n}} \\ $$

Question Number 60670 Answers: 1 Comments: 2

Question Number 60808 Answers: 1 Comments: 8

Prove or disprove that there is a positive integer suitable for n^3 + 1 ∣ n! ( n! is divided by n^3 + 1 ) n ∈ Z^+

$${Prove}\:\:{or}\:\:{disprove}\:\:{that}\:\:{there}\:\:{is} \\ $$$${a}\:\:{positive}\:\:{integer}\:\:{suitable}\:\:{for} \\ $$$$\:\:\:\:\:\:\:{n}^{\mathrm{3}} \:+\:\mathrm{1}\:\:\:\mid\:\:\:{n}!\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\:\:{n}!\:\:\:{is}\:\:{divided}\:\:{by}\:\:{n}^{\mathrm{3}} \:+\:\mathrm{1}\:\:\right) \\ $$$${n}\:\:\in\:\:\mathbb{Z}^{+} \\ $$

Question Number 60675 Answers: 0 Comments: 0

∫_0 ^(π/2) ln[((ln^2 (sin(x)))/(π^2 +ln^2 (sinx)))]((ln(cos(x)))/(tan(x)))dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left[\frac{{ln}^{\mathrm{2}} \left({sin}\left({x}\right)\right)}{\pi^{\mathrm{2}} +{ln}^{\mathrm{2}} \left({sinx}\right)}\right]\frac{{ln}\left({cos}\left({x}\right)\right)}{{tan}\left({x}\right)}{dx} \\ $$

Question Number 60662 Answers: 0 Comments: 0

Question Number 60659 Answers: 1 Comments: 1

find ∫_0 ^1 ln(x)ln(1−x^2 )dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 60658 Answers: 0 Comments: 1

calculate ∫_0 ^1 ln(x)ln(1−x)ln(1−x^2 )dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 60644 Answers: 0 Comments: 0

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