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IntegrationQuestion and Answers: Page 299

Question Number 31101 Answers: 0 Comments: 0

let give f(x)= ∫_0 ^x t^2 e^(−2t^2 ) sin(2(x−t))dt calculate f^(′′) +4f then finf f(x).

$${let}\:{give}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:{t}^{\mathrm{2}} \:{e}^{−\mathrm{2}{t}^{\mathrm{2}} } {sin}\left(\mathrm{2}\left({x}−{t}\right)\right){dt}\:{calculate} \\ $$$${f}^{''} \:+\mathrm{4}{f}\:\:{then}\:{finf}\:{f}\left({x}\right). \\ $$

Question Number 31100 Answers: 0 Comments: 1

find ∫_0 ^∞ ((cosx −cos(3x))/x) e^(−2x) dx.

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cosx}\:−{cos}\left(\mathrm{3}{x}\right)}{{x}}\:{e}^{−\mathrm{2}{x}} {dx}. \\ $$

Question Number 31098 Answers: 0 Comments: 2

find the value of ∫_1 ^∞ ((arctan(x+1) −arctanx)/x^2 )dx.

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

Question Number 31097 Answers: 0 Comments: 1

calculate interms of a and b the integral ∫_0 ^∞ ((arctan(bt) −arctan(at))/t)dt with a and b>0.

$${calculate}\:{interms}\:{of}\:{a}\:{and}\:{b}\:{the}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({bt}\right)\:−{arctan}\left({at}\right)}{{t}}{dt}\:\:{with}\:{a}\:{and}\:{b}>\mathrm{0}. \\ $$

Question Number 31096 Answers: 0 Comments: 1

find ∫_0 ^π (dx/((a+bcosx)^2 )) with a>b>0 then give the value of ∫_0 ^π (dx/((2+cosx)^2 ))

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{dx}}{\left({a}+{bcosx}\right)^{\mathrm{2}} }\:{with}\:{a}>{b}>\mathrm{0}\:{then}\:{give}\:{the} \\ $$$${value}\:{of}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\left(\mathrm{2}+{cosx}\right)^{\mathrm{2}} } \\ $$

Question Number 31095 Answers: 0 Comments: 1

find I_n (x)= ∫_0 ^∞ t^n e^(−xt) dt x>0 n∈ N.

$${find}\:{I}_{{n}} \left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{t}^{{n}} \:{e}^{−{xt}} {dt}\:\:\:\:{x}>\mathrm{0}\:{n}\in\:{N}. \\ $$

Question Number 31094 Answers: 0 Comments: 0

m and n integrs and y≥0 find ∫_0 ^y x^m (y−x)^n dx

$${m}\:{and}\:{n}\:{integrs}\:{and}\:{y}\geqslant\mathrm{0}\:{find}\:\int_{\mathrm{0}} ^{{y}} \:{x}^{{m}} \left({y}−{x}\right)^{{n}} {dx} \\ $$

Question Number 31093 Answers: 0 Comments: 1

calculate ∫_0 ^∞ e^(−x^2 ) cos(2xy)dx.

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {cos}\left(\mathrm{2}{xy}\right){dx}. \\ $$

Question Number 31092 Answers: 0 Comments: 1

find ∫_0 ^∞ ((ln(1+4x^2 ))/(1+2x^2 ))dx .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 31091 Answers: 0 Comments: 1

let −1<t<1 find f(t)= ∫_0 ^π ((ln(1+tcosx))/(cosx))dx

$${let}\:\:−\mathrm{1}<{t}<\mathrm{1}\:{find}\:{f}\left({t}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{ln}\left(\mathrm{1}+{tcosx}\right)}{{cosx}}{dx} \\ $$

Question Number 31090 Answers: 0 Comments: 1

find ∫∫_(1≤x^2 +y^2 ≤4 and y≥0) ((dxdy)/(√(x^2 +y^2 ))) .

$${find}\:\int\int_{\mathrm{1}\leqslant{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{4}\:{and}\:{y}\geqslant\mathrm{0}} \:\:\:\frac{{dxdy}}{\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}\:. \\ $$

Question Number 31089 Answers: 0 Comments: 0

find ∫_0 ^1 dy ∫_y^2 ^y ((ydx)/(x(√(x^2 +y^2 )))) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{dy}\:\int_{{y}^{\mathrm{2}} } ^{{y}} \:\:\frac{{ydx}}{{x}\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}\:. \\ $$

Question Number 31088 Answers: 0 Comments: 0

find ∫_0 ^1 dx ∫_0 ^(1−x) e^((y−x)/(y+x)) dy.

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{dx}\:\int_{\mathrm{0}} ^{\mathrm{1}−{x}} \:\:{e}^{\frac{{y}−{x}}{{y}+{x}}} \:{dy}. \\ $$

Question Number 31087 Answers: 0 Comments: 0

find ∫∫∫_(x^2 +y^2 +z^2 <4) (x^2 +y^2 +z^2 )dxdydz.

$${find}\:\int\int\int_{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}} \:<\mathrm{4}} \:\:\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+{z}^{\mathrm{2}} \right){dxdydz}. \\ $$

Question Number 31086 Answers: 0 Comments: 0

find ∫∫_D (x^4 −y^4 )dxdy with D= {(x,y)∈R^2 / 1<x^2 −y^2 <2 ,1<xy<2 ,x>0,y>0}

$${find}\:\int\int_{{D}} \left({x}^{\mathrm{4}} \:−{y}^{\mathrm{4}} \right){dxdy}\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{1}<{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} <\mathrm{2}\:,\mathrm{1}<{xy}<\mathrm{2}\:,{x}>\mathrm{0},{y}>\mathrm{0}\right\} \\ $$

Question Number 31084 Answers: 0 Comments: 1

find ∫∫_D ((dxdy)/((x+y)^4 )) with D={(x,y)∈R^2 /x≥1,y≥1,x+y≤4}

$${find}\:\int\int_{{D}} \:\:\frac{{dxdy}}{\left({x}+{y}\right)^{\mathrm{4}} }\:\:{with}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{x}\geqslant\mathrm{1},{y}\geqslant\mathrm{1},{x}+{y}\leqslant\mathrm{4}\right\} \\ $$

Question Number 31083 Answers: 0 Comments: 1

calculate by two methods ∫_0 ^1 ∫_0 ^(π/2) ((dx dt)/(1+x^2 tan^2 t)) then find the value of ∫_0 ^(π/2) t cotant dt .

$${calculate}\:{by}\:{two}\:{methods}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}\:{dt}}{\mathrm{1}+{x}^{\mathrm{2}} {tan}^{\mathrm{2}} {t}} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{t}\:{cotant}\:{dt}\:. \\ $$$$ \\ $$

Question Number 31082 Answers: 0 Comments: 0

calculate by two methods ∫_0 ^∞ ∫_0 ^∞ ((dxdy)/((1+y)(1+x^2 y))) then find the value of ∫_0 ^∞ ((lnx)/(1−x^2 ))dx.

$${calculate}\:{by}\:{two}\:{methods}\:\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dxdy}}{\left(\mathrm{1}+{y}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} {y}\right)} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{lnx}}{\mathrm{1}−{x}^{\mathrm{2}} }{dx}. \\ $$

Question Number 31081 Answers: 0 Comments: 0

find ∫_0 ^∞ dx ∫_x ^(+∞) e^(−y^2 dy) .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} {dx}\:\int_{{x}} ^{+\infty} \:{e}^{−{y}^{\mathrm{2}} {dy}} \:\:. \\ $$

Question Number 31080 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−px) dx ∫_0 ^a ((cos(xt))/(√(a^2 −t^2 )))dt with a>0 ,p>0

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{px}} {dx}\:\int_{\mathrm{0}} ^{{a}} \:\:\frac{{cos}\left({xt}\right)}{\sqrt{{a}^{\mathrm{2}} \:−{t}^{\mathrm{2}} }}{dt}\:{with}\:{a}>\mathrm{0}\:,{p}>\mathrm{0} \\ $$

Question Number 31079 Answers: 0 Comments: 0

calculate ∫∫_(0≤x≤1 and 0≤y≤2) x^2 y e^(xy) dxdxy.

$${calculate}\:\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{2}} \:\:\:{x}^{\mathrm{2}} {y}\:{e}^{{xy}} {dxdxy}. \\ $$

Question Number 31078 Answers: 0 Comments: 0

find ∫∫_(0≤x≤3 and x≤y≤4x−x^2 ) (x^2 +2y)dxdy.

$${find}\:\int\int_{\mathrm{0}\leqslant{x}\leqslant\mathrm{3}\:{and}\:{x}\leqslant{y}\leqslant\mathrm{4}{x}−{x}^{\mathrm{2}} } \:\:\:\left({x}^{\mathrm{2}} \:+\mathrm{2}{y}\right){dxdy}. \\ $$

Question Number 31077 Answers: 0 Comments: 1

calculate ∫∫_(0<x<1and 0<y<x^2 ) (y/(√(x^2 +y^2 )))dxdy.

$${calculate}\:\int\int_{\mathrm{0}<{x}<\mathrm{1}{and}\:\mathrm{0}<{y}<{x}^{\mathrm{2}} } \:\frac{{y}}{\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{dxdy}. \\ $$

Question Number 31076 Answers: 0 Comments: 1

find ∫_0 ^((√2)/2) (dx/((2x^2 +1)(√(1+x^2 )))) .

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

Question Number 31075 Answers: 0 Comments: 0

find ∫_0 ^π ((sinθ)/(cos^2 θ +2 sin^2 θ)) dθ .

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{sin}\theta}{{cos}^{\mathrm{2}} \theta\:+\mathrm{2}\:{sin}^{\mathrm{2}} \theta}\:{d}\theta\:. \\ $$

Question Number 31074 Answers: 0 Comments: 1

find ∫_a ^b (√((b−x)(x−a))) dx with a<b .then find ∫_1 ^(√2) (√(((√2) −x)(x−1))) dx.

$${find}\:\:\int_{{a}} ^{{b}} \:\sqrt{\left({b}−{x}\right)\left({x}−{a}\right)}\:{dx}\:{with}\:{a}<{b}\:.{then}\:{find}\: \\ $$$$\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{2}}} \sqrt{\left(\sqrt{\mathrm{2}}\:−{x}\right)\left({x}−\mathrm{1}\right)}\:{dx}. \\ $$

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