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let give I(x)= ∫_0 ^(π/2) (dt/(√(sin^2 t +x^2 cos^2 t))) and J(x)= ∫_0 ^(π/2) ((cost)/(√(sin^2 t +x^2 cos^2 t)))dt cslculate lim_(x→0^+ ) (I(x)−J(x)) and prove that I(x)=_(x→0^+ ) −lnx +2ln2 +o(1).

$${let}\:{give}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} \:{cos}^{\mathrm{2}} {t}}}\:\:{and} \\ $$$${J}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cost}}{\sqrt{{sin}^{\mathrm{2}} {t}\:+{x}^{\mathrm{2}} {cos}^{\mathrm{2}} {t}}}{dt}\:{cslculate}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \left({I}\left({x}\right)−{J}\left({x}\right)\right) \\ $$$${and}\:{prove}\:{that}\:\:{I}\left({x}\right)=_{{x}\rightarrow\mathrm{0}^{+} } \:−{lnx}\:+\mathrm{2}{ln}\mathrm{2}\:+{o}\left(\mathrm{1}\right). \\ $$

Question Number 28611 Answers: 0 Comments: 1

let give θ∈]0,π[ prove that ∫_0 ^1 (dt/(e^(−iθ) −t))= Σ_(n=1) ^(+∞) (e^(inθ) /n) .

$$\left.{let}\:{give}\:\theta\in\right]\mathrm{0},\pi\left[\:\:{prove}\:{that}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dt}}{{e}^{−{i}\theta} −{t}}=\:\sum_{{n}=\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{{in}\theta} }{{n}}\:\:.\right. \\ $$

Question Number 28543 Answers: 0 Comments: 0

find the value of ∫_(−∞) ^(+∞) (dt/((t^2 +t +h^2 )^2 +h^2 )) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} +{t}\:+{h}^{\mathrm{2}} \right)^{\mathrm{2}} \:+{h}^{\mathrm{2}} }\:\:\:. \\ $$

Question Number 28541 Answers: 0 Comments: 0

prove that ∫_0 ^∞ ((sinx)/(e^(ax) −1))dx= Σ_(p=1) ^∞ (1/(1+p^2 a^2 )) with a>0

$${prove}\:{that}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sinx}}{{e}^{{ax}} −\mathrm{1}}{dx}=\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{\mathrm{1}+{p}^{\mathrm{2}} {a}^{\mathrm{2}} }\:\:\:\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 28539 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((1−cos(xt))/t^2 ) e^(−t) dt .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}−{cos}\left({xt}\right)}{{t}^{\mathrm{2}} }\:{e}^{−{t}} {dt}\:. \\ $$

Question Number 28540 Answers: 0 Comments: 0

find F( e^(−ax^2 ) ) where F mean fourier transform. a>0

$${find}\:\boldsymbol{{F}}\left(\:{e}^{−{ax}^{\mathrm{2}} } \right)\:\:\:{where}\:\:\boldsymbol{{F}}\:\:{mean}\:{fourier}\:{transform}. \\ $$$${a}>\mathrm{0} \\ $$

Question Number 28448 Answers: 1 Comments: 0

find ∫∫_D ((xy)/(1+x^2 +y^2 ))dxdy with D= {(x,y)∈R^2 / x^2 +y^2 ≥1 } .

$${find}\:\int\int_{{D}} \:\:\:\:\frac{{xy}}{\mathrm{1}+{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{dxdy}\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \geqslant\mathrm{1}\:\:\right\}\:\:. \\ $$

Question Number 28446 Answers: 0 Comments: 0

find ∫∫_A (x+y) e^(−x) e^(−y) dxdy with A= {(x,y)∈R^2 /x≥0 ,y≥0 , x+y ≤1 } .

$${find}\:\int\int_{{A}} \left({x}+{y}\right)\:{e}^{−{x}} \:{e}^{−{y}} \:{dxdy}\:{with} \\ $$$${A}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:\:/{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\:,\:{x}+{y}\:\leqslant\mathrm{1}\:\right\}\:\:. \\ $$

Question Number 28445 Answers: 0 Comments: 0

find ∫∫_(x≤x^2 +y^2 ≤1) ((dxdy)/((1+x^2 +y^2 ))) .

$${find}\:\int\int_{{x}\leqslant{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:\leqslant\mathrm{1}} \:\:\frac{{dxdy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}\:\:. \\ $$

Question Number 28444 Answers: 0 Comments: 0

let give 1<a<b and I= ∫_0 ^π ∫_a ^b (du/(x−cosu)) dt find the value of ∫_0 ^π ln(((b−cost)/(a−cost)))dt .

$${let}\:{give}\:\:\mathrm{1}<{a}<{b}\:\:{and}\:{I}=\:\int_{\mathrm{0}} ^{\pi} \:\int_{{a}} ^{{b}} \:\:\frac{{du}}{{x}−{cosu}}\:{dt}\:\:{find}\:{the} \\ $$$${value}\:{of}\:\int_{\mathrm{0}} ^{\pi} \:\:{ln}\left(\frac{{b}−{cost}}{{a}−{cost}}\right){dt}\:. \\ $$

Question Number 28442 Answers: 0 Comments: 0

let give B(x,y)= ∫_0 ^1 u^(x−1) (1−u)^(y−1) du and (beta function) and Γ(x) =∫_0 ^∞ u^(x−1) e^(−u) du (x>0)(gamma function of euler) 1) prove that Γ(x)= 2∫_0 ^∞ u^(2x−1) e^(−u^2 ) du . 2) prove that B(x,y) = ((Γ(x).Γ(y))/(Γ(x+y))) .

$${let}\:{give}\:{B}\left({x},{y}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{y}−\mathrm{1}} {du}\:\:{and}\:\left({beta}\:{function}\right) \\ $$$${and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{u}^{{x}−\mathrm{1}} \:{e}^{−{u}} \:{du}\:\:\:\:\:\left({x}>\mathrm{0}\right)\left({gamma}\:{function}\:{of}\:{euler}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\:\Gamma\left({x}\right)=\:\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:{u}^{\mathrm{2}{x}−\mathrm{1}} \:{e}^{−{u}^{\mathrm{2}} \:} {du}\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:{B}\left({x},{y}\right)\:=\:\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)}\:. \\ $$$$ \\ $$

Question Number 28439 Answers: 0 Comments: 0

find ∫ ((1+tanx)/(1+sin^2 x))dx

$${find}\:\:\int\:\:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{sin}^{\mathrm{2}} {x}}{dx} \\ $$

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