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IntegrationQuestion and Answers: Page 313

Question Number 27815    Answers: 1   Comments: 0

∫((cos x−cos 2x)/(1−cos x))dx

$$\int\frac{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{2x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}\mathrm{dx} \\ $$

Question Number 27805    Answers: 0   Comments: 1

find ∫_1 ^∝ ((arctan(αx))/x^2 ) .

$${find}\:\:\int_{\mathrm{1}} ^{\propto} \:\:\frac{{arctan}\left(\alpha{x}\right)}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 27804    Answers: 0   Comments: 1

calculate ∫_0 ^∝ ((e^(−ax) − e^(−bx) )/x^2 )dx with a>0 b>o

$${calculate}\:\:\int_{\mathrm{0}} ^{\propto} \:\:\frac{{e}^{−{ax}} \:−\:{e}^{−{bx}} }{{x}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$${b}>{o} \\ $$

Question Number 27803    Answers: 0   Comments: 0

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

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

Question Number 27802    Answers: 0   Comments: 1

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

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

Question Number 27797    Answers: 0   Comments: 1

find ∫ (√(2+tan^2 t)) dt.

$${find}\:\:\:\int\:\sqrt{\mathrm{2}+{tan}^{\mathrm{2}} {t}}\:\:{dt}. \\ $$

Question Number 27796    Answers: 0   Comments: 0

find ∫ (x^2 /((cosx +x sinx)^2 )) .

$${find}\:\:\int\:\:\:\frac{{x}^{\mathrm{2}} }{\left({cosx}\:+{x}\:{sinx}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 27794    Answers: 0   Comments: 0

let give I(x)= ∫_0 ^π ln (1−2x cost +x^2 )dt by using the polynomial p(x)= (z+x)^(2n) −1 find the value of I(x).

$${let}\:{give}\:\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} {ln}\:\left(\mathrm{1}−\mathrm{2}{x}\:{cost}\:+{x}^{\mathrm{2}} \right){dt}\:{by}\:{using}\:{the} \\ $$$${polynomial}\:{p}\left({x}\right)=\:\left({z}+{x}\right)^{\mathrm{2}{n}} −\mathrm{1}\:\:{find}\:{the}\:{value}\:{of}\:{I}\left({x}\right). \\ $$

Question Number 27788    Answers: 0   Comments: 0

find the value of A_n = ∫_0 ^π ((sin(nt))/(sint))dt with n∈N^∗ .

$${find}\:{the}\:{value}\:{of}\:\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}\left({nt}\right)}{{sint}}{dt}\:\:{with}\:{n}\in{N}^{\ast} \:. \\ $$

Question Number 27781    Answers: 0   Comments: 1

find the value of F(x)=∫_0 ^(π/2) ((ln(1+x sin^2 t))/(sin^2 t)) dt knowing that −1<x<1 .

$${find}\:{the}\:{value}\:{of}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{x}\:{sin}^{\mathrm{2}} {t}\right)}{{sin}^{\mathrm{2}} {t}}\:{dt}\:{knowing}\:{that} \\ $$$$−\mathrm{1}<{x}<\mathrm{1}\:. \\ $$

Question Number 27764    Answers: 1   Comments: 0

∫(√(tan x))dx

$$\int\sqrt{\mathrm{tan}\:{x}}{dx} \\ $$

Question Number 27693    Answers: 1   Comments: 1

1) calculate ∫∫_(]0,1]×]0,(π/2)]) ((dxdy)/(1+(xtany)^2 )) 2) find the value of ∫_0 ^(π/2) (t/(tant))dt .

$$\left.\mathrm{1}\right)\:{calculate}\:\:\int\int_{\left.\right]\left.\mathrm{0}\left.,\left.\mathrm{1}\right]×\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]} \:\:\:\frac{{dxdy}}{\mathrm{1}+\left({xtany}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{t}}{{tant}}{dt}\:. \\ $$

Question Number 27692    Answers: 0   Comments: 1

find by two ways the value of ∫∫_([0,1]) x^y dxdxy then calculate ∫_0 ^1 ((t−1)/(lnt))dt .

$${find}\:{by}\:{two}\:{ways}\:{the}\:{value}\:{of}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]} \:\:{x}^{{y}} \:\:{dxdxy}\:{then} \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{t}−\mathrm{1}}{{lnt}}{dt}\:\:. \\ $$

Question Number 27691    Answers: 0   Comments: 1

let give A=∫∫_(0≤y≤x≤1) ((dxdxy)/((1+x^2 )(1+y^2 ))) and B= ∫_0 ^(π/4) ((ln(2cos^2 θ))/(2cos(2θ)))dθ calculate A and prove that B=A.

$${let}\:{give}\:\:{A}=\int\int_{\mathrm{0}\leqslant{y}\leqslant{x}\leqslant\mathrm{1}} \:\:\:\:\:\:\frac{{dxdxy}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}\:\:{and} \\ $$$${B}=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{ln}\left(\mathrm{2}{cos}^{\mathrm{2}} \theta\right)}{\mathrm{2}{cos}\left(\mathrm{2}\theta\right)}{d}\theta\:\:{calculate}\:{A}\:{and}\:{prove}\:{that}\:{B}={A}. \\ $$

Question Number 27690    Answers: 0   Comments: 1

find I= ∫∫_D ln(1+x+y)dxdy with D= {(x,y)∈R^2 / x+y≤1 and x≥0 and y≥0 }.

$${find}\:\:\:{I}=\:\:\int\int_{{D}} {ln}\left(\mathrm{1}+{x}+{y}\right){dxdy}\:\:{with} \\ $$$${D}=\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:\:\:/\:\:{x}+{y}\leqslant\mathrm{1}\:{and}\:{x}\geqslant\mathrm{0}\:{and}\:{y}\geqslant\mathrm{0}\:\right\}. \\ $$

Question Number 27757    Answers: 1   Comments: 0

calculate I= ∫_0 ^(π/2) (dx/(1+cosx)) and J= ∫_0^ ^(π/2) ((cosx)/(1+cosx))dx .

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

Question Number 28038    Answers: 0   Comments: 0

let give f(x)=(√(x+y)) +1 and D={(x,y)∈R^2 / 0≤x≤1 and −1≤y≤1} find the value of ∫∫ f(x,y)dxdy .

$${let}\:{give}\:{f}\left({x}\right)=\sqrt{{x}+{y}}\:+\mathrm{1}\:\:{and}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:\right. \\ $$$$\left.{and}\:−\mathrm{1}\leqslant{y}\leqslant\mathrm{1}\right\}\:\:{find}\:{the}\:{value}\:{of}\:\:\int\int\:{f}\left({x},{y}\right){dxdy}\:. \\ $$

Question Number 27684    Answers: 0   Comments: 1

1) prove the existence of the integral I=∫_0 ^(π/2) ((ln(1+cosx))/(cosx))dx 2)prove that I= ∫∫_D ((siny)/(1+cosx cosy))dxdy with D=[0,(π/2)]^2 3)find the value of I.

$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{the}\:{integral} \\ $$$${I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{ln}\left(\mathrm{1}+{cosx}\right)}{{cosx}}{dx} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{I}=\:\int\int_{{D}} \:\:\frac{{siny}}{\mathrm{1}+{cosx}\:{cosy}}{dxdy}\:{with}\: \\ $$$${D}=\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right]^{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{I}. \\ $$

Question Number 27666    Answers: 0   Comments: 0

let give I_n = ∫_0 ^1 (x^n /(1+x^n ))dx (1) prove that lim_(n−>∝) I_n =0 (2)calculate I_n +I_(n+1) (3) find Σ_(n=1) ^∝ (((−1)^(n−1) )/n) .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }{dx} \\ $$$$\left(\mathrm{1}\right)\:{prove}\:{that}\:\:{lim}_{{n}−>\propto} {I}_{{n}} =\mathrm{0} \\ $$$$\left(\mathrm{2}\right){calculate}\:{I}_{{n}} \:+{I}_{{n}+\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:{find}\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}}\:. \\ $$

Question Number 28200    Answers: 0   Comments: 1

let give I= ∫_0 ^1 ((ln(1+x))/(1+x^2 ))dx and J=∫∫_([0,1]^2 ) (x/((1+x^2 )(1+xy)))dxdy calculate J by two methods then find the value of I.

$${let}\:{give}\:{I}=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:{and}\:{J}=\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\frac{{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{xy}\right)}{dxdy} \\ $$$${calculate}\:{J}\:{by}\:{two}\:{methods}\:{then}\:{find}\:{the}\:{value}\:{of}\:{I}. \\ $$

Question Number 27643    Answers: 2   Comments: 1

Question Number 27621    Answers: 0   Comments: 0

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

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

Question Number 27620    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ (((−1)^x^2 )/(3+x^2 ))dx .

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

Question Number 27619    Answers: 0   Comments: 1

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

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

Question Number 27616    Answers: 0   Comments: 1

find ∫_0 ^1 e^(−2x) ln(1+x)dx .

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

Question Number 27615    Answers: 0   Comments: 2

∫x^(5/2) (1−x)^(3/2) dx

$$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$

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