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IntegrationQuestion and Answers: Page 303

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.

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

Question Number 31073 Answers: 1 Comments: 1

find I= ∫_0 ^(π/2) ((1−sinθ)/(cosθ))dθ .

$${find}\:{I}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{\mathrm{1}−{sin}\theta}{{cos}\theta}{d}\theta\:. \\ $$

Question Number 31072 Answers: 0 Comments: 0

find ∫_0 ^∞ (dx/(e^x (√(sh(2x))))) dx.

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

Question Number 31071 Answers: 1 Comments: 3

find ∫_0 ^π (dx/(1+sin^2 x)) .

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

Question Number 31070 Answers: 0 Comments: 1

calculate ∫_0 ^π (dx/(1+2cosx)) .

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}{cosx}}\:. \\ $$

Question Number 31069 Answers: 1 Comments: 1

clculate ∫_0 ^1 x(√(x^2 −2x+2)) dx

$${clculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\sqrt{{x}^{\mathrm{2}} \:−\mathrm{2}{x}+\mathrm{2}}\:{dx} \\ $$

Question Number 31068 Answers: 0 Comments: 0

find I_n =∫_(−(π/2)) ^(π/2) e^(−ax) cos^(2n) xdx .

$${find}\:\:{I}_{{n}} =\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \:{e}^{−{ax}} \:{cos}^{\mathrm{2}{n}} {xdx}\:\:. \\ $$

Question Number 31067 Answers: 0 Comments: 0

find A_n =∫_0 ^∞ x^(2n) e^(−ax^2 ) dx.

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{x}^{\mathrm{2}{n}} \:{e}^{−{ax}^{\mathrm{2}} } {dx}. \\ $$

Question Number 31066 Answers: 0 Comments: 0

find I_n =∫_0 ^(π/2) cos^(2n+1) xdx.

$${find}\:\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}^{\mathrm{2}{n}+\mathrm{1}} {xdx}. \\ $$

Question Number 31065 Answers: 0 Comments: 0

find ∫_0 ^π ((xsinx)/((1−acosx)^2 )) dx with ∣a∣<1.

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{xsinx}}{\left(\mathrm{1}−{acosx}\right)^{\mathrm{2}} }\:{dx}\:{with}\:\:\mid{a}\mid<\mathrm{1}. \\ $$

Question Number 31063 Answers: 0 Comments: 0

find f(t)= ∫_0 ^1 ln(1+tx^2 )dxfor t>−1

$${find}\:{f}\left({t}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right){dxfor}\:\:{t}>−\mathrm{1} \\ $$

Question Number 31062 Answers: 0 Comments: 0

find ∫_0 ^(π/2) e^x sinx cos^2 xdx.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{e}^{{x}} \:{sinx}\:{cos}^{\mathrm{2}} {xdx}. \\ $$

Question Number 31061 Answers: 0 Comments: 0

find ∫_0 ^(π/2) (sinθ −cosθ)ln(sinθ+cosθ)dθ.

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\left({sin}\theta\:−{cos}\theta\right){ln}\left({sin}\theta+{cos}\theta\right){d}\theta. \\ $$

Question Number 31060 Answers: 0 Comments: 0

calculate by recurrence ∫_0 ^∞ ((lnx)/((1+x)^n ))dx with n≥2 .

$${calculate}\:{by}\:{recurrence}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{lnx}}{\left(\mathrm{1}+{x}\right)^{{n}} }{dx}\:{with}\:{n}\geqslant\mathrm{2}\:. \\ $$

Question Number 31059 Answers: 0 Comments: 0

find ∫_0 ^(π/2) cos(2θ)ln(tanθ)dθ.

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left(\mathrm{2}\theta\right){ln}\left({tan}\theta\right){d}\theta. \\ $$

Question Number 31058 Answers: 0 Comments: 0

find ∫_0 ^∞ ((x arctanx)/((1+x^2 )^2 ))dx

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

Question Number 31057 Answers: 0 Comments: 0

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

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

Question Number 31056 Answers: 0 Comments: 1

find ∫_1 ^(+∞) (dx/(x^2 −2xcosα +1)) with 0<α<π .

$${find}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\alpha\:+\mathrm{1}}\:\:{with}\:\mathrm{0}<\alpha<\pi\:. \\ $$

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