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Question Number 35244 Answers: 0 Comments: 1

if y=((sin^(−1) x)/(1−x^2 )) show that (1−x^2 )(dy/dx) −xy=1

$${if}\:\:{y}=\frac{{sin}^{−\mathrm{1}} {x}}{\mathrm{1}−{x}^{\mathrm{2}} }\:\:{show}\:{that}\: \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\frac{{dy}}{{dx}}\:−{xy}=\mathrm{1} \\ $$

Question Number 35242 Answers: 1 Comments: 1

find ∫_0 ^π ((xdx)/(1+sinx))

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{xdx}}{\mathrm{1}+{sinx}} \\ $$

Question Number 35241 Answers: 2 Comments: 6

calculate ∫_0 ^π ((x dx)/(3 +cosx)) .

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{x}\:{dx}}{\mathrm{3}\:+{cosx}}\:\:. \\ $$

Question Number 35238 Answers: 0 Comments: 1

study the convergence of ∫_1 ^(+∞) ((e^(−3x) −e^(−2x) )/x^2 )dx

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{−\mathrm{3}{x}} \:−{e}^{−\mathrm{2}{x}} }{{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 35237 Answers: 0 Comments: 1

study the convergence of ∫_0 ^∞ ((e^(−x) −e^(−x^2 ) )/x)dx .

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}} \:−{e}^{−{x}^{\mathrm{2}} } }{{x}}{dx}\:. \\ $$

Question Number 35236 Answers: 0 Comments: 0

letf(x)=arctan(1+ix) with ∣x∣<1 developp f at integr serie.

$${letf}\left({x}\right)={arctan}\left(\mathrm{1}+{ix}\right)\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$${developp}\:{f}\:\:{at}\:{integr}\:{serie}. \\ $$

Question Number 35235 Answers: 0 Comments: 2

let f(x)= e^(−2x) arctanx 1) calculate f^((n)) (x) 2) find f^((n)) (0) 3) developp f at integr serie

$${let}\:{f}\left({x}\right)=\:{e}^{−\mathrm{2}{x}} \:{arctanx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 35234 Answers: 0 Comments: 1

let f(x) =e^(−x^n ) with n fromN developp f at integr serie .

$${let}\:{f}\left({x}\right)\:={e}^{−{x}^{{n}} } \:\:\:\:\:{with}\:{n}\:{fromN} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 35232 Answers: 0 Comments: 0

what is the value of cos z and sinz if z=re^(iθ) r>0 ?

$${what}\:{is}\:{the}\:{value}\:{of}\:{cos}\:{z}\:{and}\:{sinz} \\ $$$${if}\:{z}={re}^{{i}\theta} \:\:\:\:{r}>\mathrm{0}\:\:? \\ $$

Question Number 35231 Answers: 0 Comments: 1

what is the value of cos(1+i) and cos(1−i)?

$${what}\:{is}\:{the}\:{value}\:{of}\:{cos}\left(\mathrm{1}+{i}\right)\:{and} \\ $$$${cos}\left(\mathrm{1}−{i}\right)? \\ $$

Question Number 35229 Answers: 1 Comments: 2

find the value of integral ∫_0 ^∞ e^(−(2+ia)^2 t^2 ) dt with a from R ∣a∣<1.

$${find}\:{the}\:{value}\:{of}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left(\mathrm{2}+{ia}\right)^{\mathrm{2}} {t}^{\mathrm{2}} } {dt}\:\:\:\:{with}\:{a}\:{from}\:{R}\:\:\:\:\mid{a}\mid<\mathrm{1}. \\ $$

Question Number 35228 Answers: 0 Comments: 2

find the value of integral ∫_0 ^∞ e^(−px) ((sin(qx))/(√x))dx with p>0 and q>0

$${find}\:{the}\:{value}\:{of}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{px}} \:\:\:\frac{{sin}\left({qx}\right)}{\sqrt{{x}}}{dx}\:\:{with}\:{p}>\mathrm{0}\:{and}\:{q}>\mathrm{0} \\ $$

Question Number 35226 Answers: 0 Comments: 4

1) calculate f(a) = ∫_0 ^π (dx/(a sin^2 x +cos^2 x)) with a>0 2) find the value of g(a) = ∫_0 ^π ((sin^2 x)/((a sin^2 x +cos^2 x)^2 ))dx

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\:\frac{{dx}}{{a}\:{sin}^{\mathrm{2}} {x}\:\:+{cos}^{\mathrm{2}} {x}} \\ $$$${with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{sin}^{\mathrm{2}} {x}}{\left({a}\:{sin}^{\mathrm{2}} {x}\:+{cos}^{\mathrm{2}} {x}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 35225 Answers: 0 Comments: 4

1) find f(a) = ∫_0 ^(2π) (dt/(a cos^2 t + sin^2 t)) with a≠0 2) find g(a) = ∫_0 ^(2π) ((cos^2 t)/((a cos^2 t +sin^2 t)^2 ))dt

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{dt}}{{a}\:{cos}^{\mathrm{2}} {t}\:+\:{sin}^{\mathrm{2}} {t}}\:{with}\:{a}\neq\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}^{\mathrm{2}} {t}}{\left({a}\:{cos}^{\mathrm{2}} {t}\:+{sin}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }{dt}\: \\ $$

Question Number 35224 Answers: 1 Comments: 0

calculate ∫_0 ^(2π) ((1+2cost)/(5+4cost))dt

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{\mathrm{1}+\mathrm{2}{cost}}{\mathrm{5}+\mathrm{4}{cost}}{dt} \\ $$

Question Number 35223 Answers: 1 Comments: 0

what is the value of cos(i+j) with i^2 =−1 and j =e^(i((2π)/3)) ?

$${what}\:{is}\:{the}\:{value}\:{of}\:{cos}\left({i}+{j}\right)\:{with}\:{i}^{\mathrm{2}} =−\mathrm{1}\:{and} \\ $$$${j}\:={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:? \\ $$

Question Number 35222 Answers: 1 Comments: 1

let ∣x∣<1 prove that arctanx =(i/2)ln(((i+x)/(i−x)))

$${let}\:\mid{x}\mid<\mathrm{1}\:{prove}\:{that}\: \\ $$$${arctanx}\:=\frac{{i}}{\mathrm{2}}{ln}\left(\frac{{i}+{x}}{{i}−{x}}\right) \\ $$

Question Number 35221 Answers: 0 Comments: 0

let z from C prove that e^z = Σ_(n=0) ^∞ (z^n /(n!)) .

$${let}\:{z}\:{from}\:{C}\:{prove}\:{that} \\ $$$${e}^{{z}} \:=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{z}^{{n}} }{{n}!}\:. \\ $$

Question Number 35220 Answers: 0 Comments: 1

let z from C and f(z)= ((2z)/((z−1)(2z +1))) developp f at integr serie.

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

Question Number 35219 Answers: 0 Comments: 0

let z ∈C prove that cosz =ch(iz) and sinz=sh(iz)

$${let}\:{z}\:\in{C}\:\:{prove}\:{that} \\ $$$${cosz}\:={ch}\left({iz}\right)\:{and}\:{sinz}={sh}\left({iz}\right) \\ $$

Question Number 35218 Answers: 0 Comments: 0

prove that ∫_0 ^∞ (t^(a−1) /(1+t))dt =(π/(sin(πa))) that we know 0<a<1 .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}\:=\frac{\pi}{{sin}\left(\pi{a}\right)} \\ $$$${that}\:{we}\:{know}\:\mathrm{0}<{a}<\mathrm{1}\:. \\ $$

Question Number 35217 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((x sin(2x))/(x^2 +4))dx

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

Question Number 35216 Answers: 0 Comments: 0

study the function f_n (x)=arcos(ncosx) n≥1 integr.

$${study}\:{the}\:{function}\: \\ $$$${f}_{{n}} \left({x}\right)={arcos}\left({ncosx}\right)\:{n}\geqslant\mathrm{1}\:{integr}. \\ $$

Question Number 35215 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((cosx +cos(2x))/(x^2 +9))dx

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cosx}\:+{cos}\left(\mathrm{2}{x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$

Question Number 35214 Answers: 0 Comments: 0

let a>0 b ∈C and Re(b)>0 cslculate ∫_(−∞) ^(+∞) (e^(iax) /(x−ib))dx and ∫_(−∞) ^(+∞) (e^(iax) /(x+ib))dx

$${let}\:{a}>\mathrm{0}\:\:{b}\:\in{C}\:{and}\:{Re}\left({b}\right)>\mathrm{0} \\ $$$${cslculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{iax}} }{{x}−{ib}}{dx}\:\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{{iax}} }{{x}+{ib}}{dx} \\ $$

Question Number 35213 Answers: 0 Comments: 0

find the values of ∫_0 ^∞ cos(λx^2 )dx and ∫_0 ^∞ sin(λx^2 )dx with λ>0 . 2) find the values of ∫_0 ^∞ cos(x^2 )dx and ∫_0 ^∞ sin(x^2 )dx( integrals of fresnel)

$${find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left(\lambda{x}^{\mathrm{2}} \right){dx}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{sin}\left(\lambda{x}^{\mathrm{2}} \right){dx}\:{with}\:\lambda>\mathrm{0}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{\mathrm{2}} \right){dx}\left(\:{integrals}\:{of}\:{fresnel}\right) \\ $$

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