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IntegrationQuestion and Answers: Page 313

Question Number 30216 Answers: 0 Comments: 0

let I(x)= ∫_0 ^π (dt/(x^2 +cos^2 t)) 1) prove that I(x)= 2∫_0 ^(π/2) (dt/(x^2 +cos^2 t)) 2) find the value of I(x).

$${let}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{I}\left({x}\right)=\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\left({x}\right). \\ $$

Question Number 30215 Answers: 0 Comments: 0

let give J(x)= (1/π) ∫_0 ^π cos(xcost)dt 1) find J^′ and J^(′′) in form of integrals 2)prove that J^′ (x)=((−x)/π) ∫_0 ^π sin^2 t cos(xcost)dt and J is solution of d.e. xy^(′′) +y^′ +xy=0

$${let}\:{give}\:{J}\left({x}\right)=\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {cos}\left({xcost}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{J}^{'} \:{and}\:{J}^{''} \:{in}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{J}^{'} \left({x}\right)=\frac{−{x}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{2}} {t}\:{cos}\left({xcost}\right){dt}\:{and}\:{J}\:{is} \\ $$$${solution}\:{of}\:{d}.{e}.\:\:{xy}^{''} \:+{y}^{'} \:+{xy}=\mathrm{0} \\ $$

Question Number 30185 Answers: 0 Comments: 0

let I= ∫_0 ^(π/2) ((sinx)/(√(1+sinxcosx)))dx and J= ∫_0 ^(π/2) ((cosx)/(√(1+sinx cosx))) dx 1) calculate I +J 2) find I and J.

$${let}\:\:{I}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{sinx}}{\sqrt{\mathrm{1}+{sinxcosx}}}{dx}\:{and} \\ $$$${J}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cosx}}{\sqrt{\mathrm{1}+{sinx}\:{cosx}}}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}\:+{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{I}\:{and}\:{J}. \\ $$

Question Number 30184 Answers: 0 Comments: 1

find ∫_(1/2) ^2 (1+(1/x^2 ))arctanx dx . (arctan=tan^(−1) ).

$${find}\:\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{2}} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx}\:.\:\left({arctan}={tan}^{−\mathrm{1}} \right). \\ $$

Question Number 30182 Answers: 0 Comments: 2

find ∫_2 ^3 ((√(x+1))/(x(√(1−x))))dx .

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

Question Number 30181 Answers: 0 Comments: 0

find ∫ (dx/(1+x^3 +x^6 )) .

$${find}\:\:\int\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{3}} \:+{x}^{\mathrm{6}} }\:. \\ $$

Question Number 30180 Answers: 0 Comments: 1

find ∫_0 ^(π/2) ((x sinx cosx)/(tan^2 x +cotan^2 x))dx .(use the ch.x=(π/2) −t).

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{x}\:{sinx}\:{cosx}}{{tan}^{\mathrm{2}} {x}\:+{cotan}^{\mathrm{2}} {x}}{dx}\:.\left({use}\:{the}\:{ch}.{x}=\frac{\pi}{\mathrm{2}}\:−{t}\right). \\ $$

Question Number 30179 Answers: 0 Comments: 1

find ∫ (dt/(1+cost +sint)) .

$${find}\:\:\int\:\:\:\:\frac{{dt}}{\mathrm{1}+{cost}\:+{sint}}\:\:. \\ $$

Question Number 30178 Answers: 0 Comments: 1

calculate ∫_0 ^(π/2) (dx/(1+cosx cosθ)) with −π<θ<π .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{cosx}\:{cos}\theta}\:\:{with}\:−\pi<\theta<\pi\:. \\ $$

Question Number 30008 Answers: 0 Comments: 1

integrate w.r.t x ∫(e^x^2 )dx

$${integrate}\:{w}.{r}.{t}\:{x} \\ $$$$\int\left({e}^{{x}^{\mathrm{2}} } \right){dx} \\ $$

Question Number 29980 Answers: 0 Comments: 0

prove that γ= Σ_(n=1) ^∞ ((1/n) −ln(1 +(1/n))) 2)show that γ= Σ_(k=2) ^∞ (((−1)^k )/k) ξ(k).

$${prove}\:{that}\:\gamma=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\left(\frac{\mathrm{1}}{{n}}\:\:−{ln}\left(\mathrm{1}\:+\frac{\mathrm{1}}{{n}}\right)\right) \\ $$$$\left.\mathrm{2}\right){show}\:{that}\:\gamma=\:\sum_{{k}=\mathrm{2}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:\xi\left({k}\right). \\ $$

Question Number 29976 Answers: 0 Comments: 0

prove that ln(Γ(x))= −lnx −γx +Σ_(n=1) ^∞ ( (x/n) −ln( 1+(x/n))) with x>0

$${prove}\:{that} \\ $$$${ln}\left(\Gamma\left({x}\right)\right)=\:−{lnx}\:−\gamma{x}\:\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\left(\:\:\frac{{x}}{{n}}\:\:−{ln}\left(\:\mathrm{1}+\frac{{x}}{{n}}\right)\right)\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 29975 Answers: 0 Comments: 2

let give 0<α<1 1) prove that π coth(πα) −(1/α) = Σ_(n=1) ^∞ ((2α)/(α^2 +n^2 )). 2)by integration on[0,1] find Π_(n=1) ^∞ (1+(1/n^2 )).

$$\:{let}\:{give}\:\mathrm{0}<\alpha<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\pi\:{coth}\left(\pi\alpha\right)\:−\frac{\mathrm{1}}{\alpha}\:=\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{\mathrm{2}\alpha}{\alpha^{\mathrm{2}} \:+{n}^{\mathrm{2}} }. \\ $$$$\left.\mathrm{2}\right){by}\:{integration}\:{on}\left[\mathrm{0},\mathrm{1}\right]\:{find}\:\prod_{{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right). \\ $$

Question Number 29972 Answers: 0 Comments: 1

let give ∣x∣<1 find ∫_0 ^(π/2) (dθ/(√(1−x^2 cos^2 θ))) .

$${let}\:{give}\:\mid{x}\mid<\mathrm{1}\:{find}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{d}\theta}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} {cos}^{\mathrm{2}} \theta}}\:. \\ $$

Question Number 29971 Answers: 0 Comments: 2

find J(x)= ∫_0 ^∞ (dt/(x+e^t )) ?.

$${find}\:{J}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{x}+{e}^{{t}} }\:\:\:\:?. \\ $$

Question Number 29957 Answers: 1 Comments: 0

∫3xdx

$$\int\mathrm{3}{x}\mathrm{d}{x} \\ $$

Question Number 29857 Answers: 0 Comments: 0

find ∫_0 ^(+∞) ((ln(x))/((1+x)^3 ))dx .

$${find}\:\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{3}} }{dx}\:. \\ $$

Question Number 29856 Answers: 0 Comments: 1

find ∫_0 ^(2π) ((cos(nθ))/(2+3cosθ))dθ . n from N.

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\left({n}\theta\right)}{\mathrm{2}+\mathrm{3}{cos}\theta}{d}\theta\:.\:\:{n}\:{from}\:{N}. \\ $$

Question Number 29855 Answers: 1 Comments: 1

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

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{3}+{x}^{\mathrm{2}} \right)}{dx}\:. \\ $$

Question Number 29854 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) (((x^2 +2)dx)/(x^4 +8x^2 −16x +20)) .

$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\:\frac{\left({x}^{\mathrm{2}} +\mathrm{2}\right){dx}}{{x}^{\mathrm{4}} \:+\mathrm{8}{x}^{\mathrm{2}} −\mathrm{16}{x}\:+\mathrm{20}}\:. \\ $$

Question Number 29853 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) (dx/(x^2 +2ix +2−4i)) .

$${find}\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} +\mathrm{2}{ix}\:+\mathrm{2}−\mathrm{4}{i}}\:. \\ $$

Question Number 29852 Answers: 0 Comments: 0

let f(z) =z cos^2 ((π/z)) find Res(f,0).

$${let}\:{f}\left({z}\right)\:={z}\:{cos}^{\mathrm{2}} \left(\frac{\pi}{{z}}\right)\:\:{find}\:{Res}\left({f},\mathrm{0}\right). \\ $$

Question Number 29850 Answers: 0 Comments: 0

find I = ∫_0 ^∞ (((1+x)^(−(1/4)) −(1+x)^(−(3/4)) )/x)dx .

$${find}\:\:{I}\:\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(\mathrm{1}+{x}\right)^{−\frac{\mathrm{1}}{\mathrm{4}}} \:\:−\left(\mathrm{1}+{x}\right)^{−\frac{\mathrm{3}}{\mathrm{4}}} }{{x}}{dx}\:. \\ $$

Question Number 29849 Answers: 0 Comments: 1

let give a>0 ,b>0 find the vslue of ∫_0 ^(+∞) ((e^(−at) −e^(−bt) )/t) cos(xt)dt .

$${let}\:{give}\:{a}>\mathrm{0}\:,{b}>\mathrm{0}\:{find}\:{the}\:{vslue}\:{of}\: \\ $$$$\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{at}} \:−{e}^{−{bt}} }{{t}}\:{cos}\left({xt}\right){dt}\:. \\ $$

Question Number 29574 Answers: 0 Comments: 2

∫x^6 −1/x^2 −1dx

$$\int{x}^{\mathrm{6}} −\mathrm{1}/{x}^{\mathrm{2}} −\mathrm{1}{dx} \\ $$

Question Number 29553 Answers: 0 Comments: 0

let put I(x)= ∫_x ^(+∞) ((sin^3 t)/t^2 )dt with x>0 find lim_(x→0^+ ) I(x) 2) find ∫_0 ^∞ ((sin^3 t)/t^2 )dt .

$${let}\:{put}\:\:{I}\left({x}\right)=\:\int_{{x}} ^{+\infty} \:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$${find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{I}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}^{\mathrm{3}} {t}}{{t}^{\mathrm{2}} }{dt}\:. \\ $$

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