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Question Number 34314 Answers: 0 Comments: 1

let f(x)= ∫_0 ^(+∞) ((1−cos(xt))/t^2 ) e^(−t) dt calculate f(x) .

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{\mathrm{1}−{cos}\left({xt}\right)}{{t}^{\mathrm{2}} }\:{e}^{−{t}} {dt}\: \\ $$$${calculate}\:{f}\left({x}\right)\:. \\ $$

Question Number 34313 Answers: 0 Comments: 0

let u_0 =x ≠o and u_(n+1) =ln(((e^u_n −1)/u_n )) 1) study the convervence of (u_n ) 2)find Σ_(n=0) ^∞ (Π_(k=0) ^n u_k ) .

$${let}\:{u}_{\mathrm{0}} ={x}\:\neq{o}\:\:{and}\:{u}_{{n}+\mathrm{1}} ={ln}\left(\frac{{e}^{{u}_{{n}} } \:−\mathrm{1}}{{u}_{{n}} }\right) \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convervence}\:{of}\:\left({u}_{{n}} \right) \\ $$$$\left.\mathrm{2}\right){find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\left(\prod_{{k}=\mathrm{0}} ^{{n}} \:{u}_{{k}} \right)\:. \\ $$

Question Number 34312 Answers: 0 Comments: 1

calculate I = ∫∫_D x^3 dxdy on the domain D ={(x,y)∈R^2 /1≤x≤2 , x^2 −y^2 −1≥0}

$${calculate}\:{I}\:\:=\:\int\int_{{D}} {x}^{\mathrm{3}} {dxdy}\:\:\:{on}\:{the}\:{domain} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{1}\leqslant{x}\leqslant\mathrm{2}\:,\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −\mathrm{1}\geqslant\mathrm{0}\right\} \\ $$

Question Number 34311 Answers: 1 Comments: 0

let give the d.e. (1+x^2 )y^(′′) +3xy^′ +y =0find a solution y(x) deveppable at integr serie with∣x∣<1 .

$${let}\:{give}\:{the}\:{d}.{e}.\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{''} \:+\mathrm{3}{xy}^{'} \:+{y}\:=\mathrm{0}{find} \\ $$$${a}\:{solution}\:{y}\left({x}\right)\:{deveppable}\:{at}\:{integr}\:{serie}\: \\ $$$${with}\mid{x}\mid<\mathrm{1}\:. \\ $$

Question Number 34310 Answers: 0 Comments: 0

let f(x)= ∫_(−∞) ^x (dt/(1+t^2 +t^4 )) 1) prove that f id derivsble and calculate f^′ (x) 2)devellpp f at integr serie at o.

$${let}\:{f}\left({x}\right)=\:\int_{−\infty} ^{{x}} \:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} \:+{t}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\:{id}\:{derivsble}\:{and}\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){devellpp}\:{f}\:{at}\:{integr}\:{serie}\:{at}\:{o}. \\ $$

Question Number 34309 Answers: 0 Comments: 3

let S(x)= Σ_(n=1) ^∞ (−1)^(n−1) (x^(2n+1) /(4n^2 −1)) 1) find the radius of convergence 2) calculate the sum S(x).

$${let}\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{4}{n}^{\mathrm{2}} \:−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{radius}\:{of}\:{convergence} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{the}\:{sum}\:\:{S}\left({x}\right). \\ $$

Question Number 34308 Answers: 0 Comments: 0

let I = ∫_0 ^(+∞) (((1+x)^(−(1/4)) −(1+x)^(−(3/4)) )/x)dx prove that I isconvergent and find its value .

$${let}\:\:{I}\:=\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{\left(\mathrm{1}+{x}\right)^{−\frac{\mathrm{1}}{\mathrm{4}}} \:\:−\left(\mathrm{1}+{x}\right)^{−\frac{\mathrm{3}}{\mathrm{4}}} }{{x}}{dx} \\ $$$${prove}\:{that}\:{I}\:{isconvergent}\:{and}\:{find}\:{its}\:{value}\:. \\ $$

Question Number 34307 Answers: 0 Comments: 1

let give A_n = (((1 (α/n))),((−(α/n) 1)) ) calculate lim_(n→+∞) A_n ^n .

$${let}\:{give}\:{A}_{{n}} =\:\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\frac{\alpha}{{n}}}\\{−\frac{\alpha}{{n}}\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} ^{{n}} \:\:\:\:. \\ $$

Question Number 34305 Answers: 0 Comments: 1

let] A = (((1 1 0)),((1 1 1)) ) (0 1 1 1)find the caractetistic polynom of A 2) calculate A^n

$$\left.{let}\right]\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\right. \\ $$$$\left.\mathrm{1}\right){find}\:{the}\:{caractetistic}\:{polynom}\:{of}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \\ $$

Question Number 34301 Answers: 1 Comments: 0

If log_(12) 18=a ,find log_(24) 16 in term of a

$${If}\:{log}_{\mathrm{12}} \mathrm{18}={a}\:,{find}\:{log}_{\mathrm{24}} \mathrm{16}\:{in}\:{term} \\ $$$${of}\:\:{a} \\ $$

Question Number 34298 Answers: 0 Comments: 2

let A_ = ∫_0 ^∞ e^(−x) cos[x]dx and B = ∫_0 ^∞ e^(−[x]) cosxdx calculate A−B .

$${let}\:{A}_{\:} =\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {cos}\left[{x}\right]{dx}\:\:{and}\:{B}\:=\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left[{x}\right]} \:{cosxdx} \\ $$$${calculate}\:{A}−{B}\:\:. \\ $$

Question Number 34297 Answers: 1 Comments: 1

find ∫_(−∞) ^(+∞) e^(−z t^2 ) dt with z=r e^(iθ) ∈ C .

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−{z}\:{t}^{\mathrm{2}} } {dt}\:\:\:{with}\:{z}={r}\:{e}^{{i}\theta} \:\:\in\:{C}\:. \\ $$

Question Number 34296 Answers: 0 Comments: 3

find ∫_(−∞) ^(+∞) e^(−jx^2 ) with j =e^(i((2π)/3))

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−{jx}^{\mathrm{2}} } \:\:\:\:{with}\:\:{j}\:={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$

Question Number 34295 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−n[x]) cos(x)dx with n>0

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{n}\left[{x}\right]} \:{cos}\left({x}\right){dx}\:\:\:{with}\:{n}>\mathrm{0}\: \\ $$

Question Number 34294 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−nx) ∣sinx∣dx with n>0

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{nx}} \mid{sinx}\mid{dx}\:\:{with}\:{n}>\mathrm{0} \\ $$

Question Number 34293 Answers: 0 Comments: 0

calculate ∫∫_D x^2 y dxdy? with D = {(x,y)∈ R^2 / 0≤y≤1−x^2 ,∣x+y +3∣ ≤5}

$${calculate}\:\int\int_{{D}} \:\:{x}^{\mathrm{2}} {y}\:{dxdy}?\:\:{with} \\ $$$${D}\:=\:\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:\mathrm{0}\leqslant{y}\leqslant\mathrm{1}−{x}^{\mathrm{2}} \:,\mid{x}+{y}\:+\mathrm{3}\mid\:\leqslant\mathrm{5}\right\} \\ $$

Question Number 34292 Answers: 0 Comments: 1

calculate ∫∫_w (x+y)e^(x−y) dxdy with w={(x,y)∈R^2 / ∣x∣ ≤1 and ∣y+1∣≤3 }

$${calculate}\:\int\int_{{w}} \:\left({x}+{y}\right){e}^{{x}−{y}} {dxdy}\:{with} \\ $$$${w}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\:\mid{x}\mid\:\leqslant\mathrm{1}\:\:{and}\:\mid{y}+\mathrm{1}\mid\leqslant\mathrm{3}\:\right\} \\ $$

Question Number 34291 Answers: 0 Comments: 0

let B(x,y) = ∫_0 ^1 u^(x−1) (1−u)^(y−1) du and Γ(x)= ∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x) = 2∫_0 ^∞ u^(2x−1) e^(−u^2 ) du 2)give Γ(x)Γ(y) at form of double integrale 3)prove that B(x,y) =((Γ(x)Γ(y))/(Γ(x+y))) 4) calculate B(m,n) for m and n integr naturals

$${let}\:{B}\left({x},{y}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} {u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{y}−\mathrm{1}} \:{du}\:\:{and} \\ $$$$\Gamma\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\infty} \:{u}^{\mathrm{2}{x}−\mathrm{1}} \:{e}^{−{u}^{\mathrm{2}} } {du} \\ $$$$\left.\mathrm{2}\right){give}\:\Gamma\left({x}\right)\Gamma\left({y}\right)\:{at}\:{form}\:{of}\:{double}\:{integrale} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{B}\left({x},{y}\right)\:=\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{B}\left({m},{n}\right)\:{for}\:{m}\:{and}\:{n}\:{integr}\:{naturals} \\ $$

Question Number 34290 Answers: 0 Comments: 0

calculate ∫∫_D ((dxdy)/((1+x+y)^2 )) D ={(x,y)∈ R^2 / 1≤x+y≤ 2}

$${calculate}\:\int\int_{{D}} \:\:\:\:\:\:\:\frac{{dxdy}}{\left(\mathrm{1}+{x}+{y}\right)^{\mathrm{2}} } \\ $$$${D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:/\:\:\mathrm{1}\leqslant{x}+{y}\leqslant\:\mathrm{2}\right\}\: \\ $$

Question Number 34289 Answers: 0 Comments: 1

calculate ∫∫_w (xy −2)dxdy with w = {(x,y)∈R^2 / x≥0 and 1≤y≤2−x }

$$\:{calculate}\:\int\int_{{w}} \:\:\left({xy}\:−\mathrm{2}\right){dxdy}\:\:{with}\: \\ $$$${w}\:=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:/\:\:{x}\geqslant\mathrm{0}\:{and}\:\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}−{x}\:\right\} \\ $$

Question Number 34288 Answers: 0 Comments: 0

calculate ∫∫_w e^(−yx^2 ) (x+y)dxdy with w =[0,1]^2

$${calculate}\:\int\int_{{w}} \:{e}^{−{yx}^{\mathrm{2}} } \left({x}+{y}\right){dxdy}\:\:{with} \\ $$$${w}\:=\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} \\ $$

Question Number 34287 Answers: 0 Comments: 0

calculate ∫∫_D xydxdy with D={(x,y)∈R^2 /x≥0 ,y≥0 , x+y ≤ (3/2)}

$${calculate}\:\int\int_{{D}} \:{xydxdy}\:{with} \\ $$$${D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{0}\:,{y}\geqslant\mathrm{0}\:,\:{x}+{y}\:\leqslant\:\frac{\mathrm{3}}{\mathrm{2}}\right\} \\ $$

Question Number 34286 Answers: 0 Comments: 2

find ∫ ((artanx)/((1+x)^2 ))dx

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

Question Number 34285 Answers: 0 Comments: 3

find ∫ (dx/((1+chx)^2 )) 2) calculate ∫_0 ^1 (dx/((1+chx)^2 ))

$${find}\:\:\int\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{chx}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{chx}\right)^{\mathrm{2}} } \\ $$

Question Number 34284 Answers: 0 Comments: 1

find ∫ (dt/(sin(2t)))

$${find}\:\:\int\:\:\:\frac{{dt}}{{sin}\left(\mathrm{2}{t}\right)} \\ $$

Question Number 34283 Answers: 0 Comments: 2

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

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\:\frac{{dx}}{{cos}^{\mathrm{4}} {x}\:+{sin}^{\mathrm{4}} {x}} \\ $$

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