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Question Number 34716    Answers: 0   Comments: 1

calculate ∫∫_w x(√(x^2 +y^2 )) dxdy w ={(x,y)/ x^2 +y^2 ≤3 }

$${calculate}\:\int\int_{{w}} {x}\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\:{dxdy} \\ $$$${w}\:=\left\{\left({x},{y}\right)/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{3}\:\right\}\: \\ $$

Question Number 34715    Answers: 0   Comments: 0

calculate ∫∫_(0≤x≤y≤1) ((dxdy)/((x^2 +1)(y^2 +3))) .

$${calculate}\:\int\int_{\mathrm{0}\leqslant{x}\leqslant{y}\leqslant\mathrm{1}} \:\:\:\frac{{dxdy}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({y}^{\mathrm{2}} \:+\mathrm{3}\right)}\:. \\ $$

Question Number 34714    Answers: 0   Comments: 1

calculate ∫∫_(x^2 +2y^2 ≤1) (x^2 −y^2 )dxdy

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

Question Number 34713    Answers: 0   Comments: 1

let a>0 calculate ∫∫_(x^2 +y^2 ≤3) (1/(2 +x^2 +y^2 ))dxdy.

$${let}\:{a}>\mathrm{0}\:\:{calculate}\:\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant\mathrm{3}} \:\frac{\mathrm{1}}{\mathrm{2}\:+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy}. \\ $$

Question Number 34675    Answers: 0   Comments: 0

provethat e = Σ_(k=0) ^n (1/(k!)) +∫_0 ^1 (((1−t)^n )/(n!)) e^t dt .

$${provethat}\:{e}\:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\mathrm{1}}{{k}!}\:\:+\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\left(\mathrm{1}−{t}\right)^{{n}} }{{n}!}\:{e}^{{t}} \:{dt}\:. \\ $$

Question Number 34674    Answers: 0   Comments: 0

find ∫_0 ^π ((x sinx)/(1+cos^2 x)) dx

$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{x}\:{sinx}}{\mathrm{1}+{cos}^{\mathrm{2}} {x}}\:{dx} \\ $$

Question Number 34662    Answers: 0   Comments: 0

calculate I(a) =∫_(1/a) ^a ((ln(x))/(1+x^2 )) dx with a>0 2) calculate ∫_0 ^(+∞) ((ln(x))/(1+x^2 )) dx .

$${calculate}\:{I}\left({a}\right)\:\:=\int_{\frac{\mathrm{1}}{{a}}} ^{{a}} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{ln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 34661    Answers: 0   Comments: 0

let f(x)= ∫_0 ^1 (e^(−(1+t^2 )x) /(1+t^2 )) dt find a simple form of f(x)

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{−\left(\mathrm{1}+{t}^{\mathrm{2}} \right){x}} }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{find}\:{a}\:{simple}\:{form}\:{of} \\ $$$${f}\left({x}\right) \\ $$

Question Number 34635    Answers: 2   Comments: 4

calculate A(α) = ∫_0 ^1 ln(1+αix)dx 2) calculate ∫_0 ^1 ln(1+ix) dx (i^2 =−1)

$${calculate}\:{A}\left(\alpha\right)\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+\alpha{ix}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{ix}\right)\:{dx}\:\:\:\:\left({i}^{\mathrm{2}} \:=−\mathrm{1}\right) \\ $$

Question Number 34633    Answers: 0   Comments: 0

let f(α) = ∫_(−∞) ^(+∞) ((arctan(1+αxi))/(1+x^2 ))dx find f(α) .

$${let}\:{f}\left(\alpha\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{1}+\alpha{xi}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$${find}\:{f}\left(\alpha\right)\:. \\ $$

Question Number 34593    Answers: 0   Comments: 0

1) calculate ∫_(−∞) ^(+∞) ((cos(αx^n ))/(x^2 +x +1)) dx with n integr natural 2) find the value of ∫_(−∞) ^∞ ((cos( α x^(2n) ))/(x^2 +x +1))dx 3) calculate ∫_(−∞) ^(+∞) ((cos(π x^3 ))/(x^2 +x +1)) dx

$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\alpha{x}^{{n}} \right)}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}\:{dx}\:\:{with}\:{n}\:{integr} \\ $$$${natural} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{\infty} \:\:\:\:\frac{{cos}\left(\:\alpha\:{x}^{\mathrm{2}{n}} \right)}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left(\pi\:{x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+{x}\:+\mathrm{1}}\:{dx} \\ $$

Question Number 34562    Answers: 1   Comments: 1

find the value of ∫_0 ^1 ((arctanx)/((1+x^2 )^2 )) dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctanx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 34561    Answers: 0   Comments: 1

find the value of ∫_0 ^(+∞) ((arctan(x))/((1+x^2 )^2 )) dx

$${find}\:{the}\:{value}\:\:{of}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{arctan}\left({x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 34985    Answers: 1   Comments: 0

Question Number 34421    Answers: 0   Comments: 1

let A = ∫_(−∞) ^(+∞) (dx/(x^2 −j)) with j=e^(i((2π)/3)) extract ReA and Im(A) and calculste its values.

$${let}\:{A}\:\:=\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−{j}}\:\:\:\:{with}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$${extract}\:\:{ReA}\:{and}\:{Im}\left({A}\right)\:{and}\:{calculste}\:{its}\:{values}. \\ $$

Question Number 34320    Answers: 0   Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/(x^2 +1 −i))

$${calculate}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\mathrm{1}\:−{i}} \\ $$

Question Number 34316    Answers: 0   Comments: 0

find a eajivalent of u_n = ∫_0 ^∞ e^(−(t/n)) arcctant dt .

$${find}\:{a}\:{eajivalent}\:{of} \\ $$$${u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:{e}^{−\frac{{t}}{{n}}} \:\:\:{arcctant}\:{dt}\:. \\ $$

Question Number 34315    Answers: 0   Comments: 2

1) find F(x)= ∫_0 ^(+∞) ((e^(−at) −e^(−bt) )/t)sin(xt)dt with a>0 ,b>0 .

$$\left.\mathrm{1}\right)\:{find}\:\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{at}} \:−{e}^{−{bt}} }{{t}}{sin}\left({xt}\right){dt} \\ $$$${with}\:{a}>\mathrm{0}\:,{b}>\mathrm{0}\:. \\ $$

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 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 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 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

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

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