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Question Number 28449 Answers: 2 Comments: 0

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 28688 Answers: 0 Comments: 0

Question Number 28439 Answers: 0 Comments: 0

find ∫ ((1+tanx)/(1+sin^2 x))dx

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

Question Number 28438 Answers: 0 Comments: 0

find I= ∫_0 ^∞ (dt/((1+t^4 )^n )) with n integr and n≠0.

$${find}\:\:{I}=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{and}\:{n}\neq\mathrm{0}. \\ $$

Question Number 28437 Answers: 0 Comments: 0

let give f(x)= sin ((π/x)) find f^((n)) .

$${let}\:{give}\:{f}\left({x}\right)=\:{sin}\:\left(\frac{\pi}{{x}}\right)\:\:\:{find}\:{f}^{\left({n}\right)} . \\ $$

Question Number 28436 Answers: 0 Comments: 0

study the nature of sequence (u_n ) / u_0 =1 and u_(n+1) = (1/(u_(n ) + e^(−n) )) .

$${study}\:{the}\:{nature}\:{of}\:{sequence}\:\left({u}_{{n}} \right)\:/\:{u}_{\mathrm{0}} =\mathrm{1}\:{and} \\ $$$${u}_{{n}+\mathrm{1}} =\:\:\frac{\mathrm{1}}{{u}_{{n}\:} +\:{e}^{−{n}} }\:. \\ $$

Question Number 28435 Answers: 0 Comments: 0

find Q(x) / nx^(n+1) −(n+1)x^n +1 =(x−1)^2 Q(x).

$${find}\:{Q}\left({x}\right)\:/\:{nx}^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right){x}^{{n}} +\mathrm{1}\:=\left({x}−\mathrm{1}\right)^{\mathrm{2}} {Q}\left({x}\right). \\ $$

Question Number 28434 Answers: 0 Comments: 0

let give the polynomial p(x)=(x+1)^n −(x−1)^n with n from N^∗ 1) give the factorisation of p(x) inside C[x] 2) prove that Π_(k=0) ^(n−1) cotan(((kπ)/(2p+1)))=(1/(√(2p+1)))

$${let}\:{give}\:{the}\:{polynomial}\:{p}\left({x}\right)=\left({x}+\mathrm{1}\right)^{{n}} −\left({x}−\mathrm{1}\right)^{{n}} {with}\:{n} \\ $$$${from}\:{N}^{\ast} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{the}\:{factorisation}\:{of}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cotan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right)=\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}+\mathrm{1}}} \\ $$

Question Number 28433 Answers: 0 Comments: 0

let give T_n (x)= cos(narcosx) decompose (1/(T_n (x))).

$${let}\:{give}\:{T}_{{n}} \left({x}\right)=\:{cos}\left({narcosx}\right)\:\:{decompose}\:\frac{\mathrm{1}}{{T}_{{n}} \left({x}\right)}. \\ $$

Question Number 28432 Answers: 0 Comments: 1

let put w=e^(i((2π)/n)) calculate S_n = Σ_(k=0) ^(n−1) (1/(x−w^k )) and W_n = Σ_(k=0) ^(n−1) (1/((x−w^k )^2 )) .

$${let}\:{put}\:{w}={e}^{{i}\frac{\mathrm{2}\pi}{{n}}} \:{calculate}\:\:{S}_{{n}} =\:\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\:\frac{\mathrm{1}}{{x}−{w}^{{k}} }\:\:{and} \\ $$$${W}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\:\frac{\mathrm{1}}{\left({x}−{w}^{{k}} \right)^{\mathrm{2}} }\:. \\ $$

Question Number 28431 Answers: 0 Comments: 0

let give w_k = e^(i((2kπ)/n)) k∈Z find the value of Π_(k=0) ^(n−1) (1+(2/(2−w_k ))).

$${let}\:{give}\:{w}_{{k}} =\:{e}^{{i}\frac{\mathrm{2}{k}\pi}{{n}}} \:\:\:{k}\in{Z}\:\:\:{find}\:{the}\:{value}\:{of} \\ $$$$\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left(\mathrm{1}+\frac{\mathrm{2}}{\mathrm{2}−{w}_{{k}} \:}\right). \\ $$

Question Number 28430 Answers: 0 Comments: 1

let give A_n = ∫_0 ^n (1+(x/n))^n e^(−2x) dx lim_(n→∝) A_n ?