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IntegrationQuestion and Answers: Page 302

Question Number 31415 Answers: 0 Comments: 0

let 0<x<1 find f(x)=∫_0 ^x lnt .ln(1−t)dt.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {lnt}\:.{ln}\left(\mathrm{1}−{t}\right){dt}. \\ $$

Question Number 31414 Answers: 0 Comments: 0

find ∫_0 ^((√3) ) (dx/(x^2 +(√(x+1)))) .

$${find}\:\:\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}\:} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:\:+\sqrt{{x}+\mathrm{1}}}\:. \\ $$

Question Number 31296 Answers: 0 Comments: 11

find ∫_0 ^(+∞) (dx/((1+x^2 )^n )) with n integr and n≥1 .

$${find}\:\:\int_{\mathrm{0}} ^{+\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1}\:. \\ $$

Question Number 31107 Answers: 0 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^n )) with n>1.

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{{n}} }\:\:{with}\:{n}>\mathrm{1}. \\ $$

Question Number 31106 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−x^2 ) =lim_(n→+∞) ∫_0 ^∞ (dx/((1+x^2 )^n )) . 2) prove that (1/(√π)) =lim_(n→∞) ((1.3.5....(2n−3))/(2.4.6....(2n−2))) (√n) (wallis formula).

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } ={lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:. \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\frac{\mathrm{1}}{\sqrt{\pi}}\:={lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}....\left(\mathrm{2}{n}−\mathrm{3}\right)}{\mathrm{2}.\mathrm{4}.\mathrm{6}....\left(\mathrm{2}{n}−\mathrm{2}\right)}\:\sqrt{{n}} \\ $$$$\left({wallis}\:{formula}\right). \\ $$

Question Number 31105 Answers: 0 Comments: 1

prove that ∫_0 ^x e^(−t^2 ) dt =((√π)/2) −(e^(−x^2 ) /(√π)) ∫_0 ^∞ (e^(−x^2 t^2 ) /(1+t^2 )) dt with x>0

$${prove}\:{that}\:\int_{\mathrm{0}} ^{{x}} \:\:\:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:−\frac{{e}^{−{x}^{\mathrm{2}} } }{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} {t}^{\mathrm{2}} } }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 31104 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) e^(−(x^2 +2x−1)) dx .

$${find}\:\:\:\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{1}\right)} {dx}\:. \\ $$

Question Number 31103 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x −(a/x))^2 ) dx with a≥0 .

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({x}\:−\frac{{a}}{{x}}\right)^{\mathrm{2}} } {dx}\:\:{with}\:\:{a}\geqslant\mathrm{0}\:. \\ $$

Question Number 31102 Answers: 0 Comments: 2

find ∫_0 ^(+∞) ((lnx)/(x^2 +a^2 ))dx 2) find the value of ∫_0 ^∞ ((lnx)/((x^2 +a^2 )^3 )) .

$${find}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{lnx}}{{x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnx}}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{3}} }\:. \\ $$

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 .

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

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