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IntegrationQuestion and Answers: Page 280

Question Number 41351 Answers: 1 Comments: 3

∫_0 ^∞ [(5/e^x )]dx=

$$\int_{\mathrm{0}} ^{\infty} \left[\frac{\mathrm{5}}{\mathrm{e}^{\mathrm{x}} }\right]\mathrm{dx}= \\ $$

Question Number 40625 Answers: 2 Comments: 0

Question Number 40624 Answers: 0 Comments: 0

let f(x)=∫_0 ^(π/2) ln(((1−xsint)/(1+xsint)))dt . 1) find the value of I = ∫_0 ^(π/2) ln(1−xsint)dt and J = ∫_0 ^(π/2) ln(1+xsint)dt 2) find a simple form of f(x) 3) developp f at integr serie

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\frac{\mathrm{1}−{xsint}}{\mathrm{1}+{xsint}}\right){dt}\:\:. \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left(\mathrm{1}−{xsint}\right){dt} \\ $$$${and}\:{J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{xsint}\right){dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 40621 Answers: 2 Comments: 0

let f(x)=∫_0 ^(π/2) ln(1+xcosθ)dθ 1) calculate f(1) 2) find a simple form of f(x) 3) developp f at ontehr serie

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{xcos}\theta\right){d}\theta\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left(\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{ontehr}\:{serie} \\ $$

Question Number 40620 Answers: 3 Comments: 0

find ∫ ((x+1)(√(1+x^2 )) +(1+x^2 )(√(x+1)))dx

$${find}\:\:\int\:\:\:\left(\left({x}+\mathrm{1}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$

Question Number 40619 Answers: 0 Comments: 2

let f(x)=∫_0 ^(π/2) (dθ/(x +cos^2 θ)) with x>0 . 1) calculate f(x) and f^′ (x) 2) find f^((n)) (x) and f^((n)) (0) 3) developp f at integr serie.

$${let}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{d}\theta}{{x}\:\:+{cos}^{\mathrm{2}} \theta}\:\:{with}\:{x}>\mathrm{0}\:. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right)\:{and}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 40580 Answers: 0 Comments: 3

find ∫_0 ^∞ ((ln(1+ix))/(x^3 +8))dx

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{ix}\right)}{{x}^{\mathrm{3}} \:+\mathrm{8}}{dx} \\ $$$$ \\ $$

Question Number 40569 Answers: 0 Comments: 0

Question Number 40505 Answers: 1 Comments: 0

calcilate ∫_(π/6) ^(π/4) ((sin(x))/(cos(x) +cos(2x)))dx

$${calcilate}\:\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{sin}\left({x}\right)}{{cos}\left({x}\right)\:+{cos}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 40442 Answers: 2 Comments: 0

Solve : y^4 dx + 2xy^3 dy = ((ydx− xdy)/(x^3 y^3 )).

$$\mathrm{Solve}\:: \\ $$$$\mathrm{y}^{\mathrm{4}} \mathrm{d}{x}\:+\:\mathrm{2}{x}\mathrm{y}^{\mathrm{3}} \mathrm{dy}\:=\:\frac{\mathrm{yd}{x}−\:{x}\mathrm{dy}}{{x}^{\mathrm{3}} \mathrm{y}^{\mathrm{3}} }. \\ $$

Question Number 40422 Answers: 1 Comments: 1

Solve: ydx − xdy +log xdx =0

$$\mathrm{Solve}: \\ $$$$\mathrm{yd}{x}\:−\:{xdy}\:+\mathrm{log}\:{xdx}\:=\mathrm{0} \\ $$

Question Number 40399 Answers: 3 Comments: 0

Solve : (2(√(xy)) −x)dy + ydx = 0.

$$\mathrm{Solve}\:: \\ $$$$\left(\mathrm{2}\sqrt{{xy}}\:−{x}\right){dy}\:+\:{ydx}\:=\:\mathrm{0}. \\ $$

Question Number 40397 Answers: 1 Comments: 0

Solve : (dy/dx) = ((sin y + x)/(sin 2y − xcos y)) .

$$\mathrm{Solve}\:: \\ $$$$\frac{\mathrm{dy}}{{dx}}\:=\:\frac{\mathrm{sin}\:{y}\:+\:{x}}{\mathrm{sin}\:\mathrm{2}{y}\:−\:{x}\mathrm{cos}\:{y}}\:. \\ $$

Question Number 40380 Answers: 2 Comments: 1

Solve : (dy/dx) = ((x+y)/(x−y))

$${S}\mathrm{olve}\::\:\:\:\:\:\frac{\mathrm{dy}}{\mathrm{d}{x}}\:=\:\frac{{x}+{y}}{{x}−{y}} \\ $$

Question Number 40467 Answers: 1 Comments: 2

∫ln ∣(√(x+1))+(√x)∣ dx=

$$\int\mathrm{ln}\:\mid\sqrt{{x}+\mathrm{1}}+\sqrt{{x}}\mid\:{dx}= \\ $$

Question Number 40322 Answers: 1 Comments: 4

Solve : (d^2 y/dx^2 ) = ((dy/dx))^2

$$\mathrm{Solve}\:: \\ $$$$\:\frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{d}{x}^{\mathrm{2}} }\:=\:\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} \\ $$

Question Number 40270 Answers: 1 Comments: 0

calculate the area of one “leaf” of r=sin nθ n∈N

$$\mathrm{calculate}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{one}\:``\mathrm{leaf}''\:\mathrm{of} \\ $$$${r}=\mathrm{sin}\:{n}\theta \\ $$$${n}\in\mathbb{N} \\ $$

Question Number 40251 Answers: 1 Comments: 0

1. ∫(dα/(sin 2α +tan 3α))=? 2. ∫(dβ/(cos 2β +cos 3β))=? 3. ∫(dγ/(sinh 2γ +tanh 3γ))=? 4. ∫(dδ/(cosh 2δ +cosh 3δ))=?

$$\mathrm{1}.\:\:\:\:\:\int\frac{{d}\alpha}{\mathrm{sin}\:\mathrm{2}\alpha\:+\mathrm{tan}\:\mathrm{3}\alpha}=? \\ $$$$\mathrm{2}.\:\:\:\:\:\int\frac{{d}\beta}{\mathrm{cos}\:\mathrm{2}\beta\:+\mathrm{cos}\:\mathrm{3}\beta}=? \\ $$$$\mathrm{3}.\:\:\:\:\:\int\frac{{d}\gamma}{\mathrm{sinh}\:\mathrm{2}\gamma\:+\mathrm{tanh}\:\mathrm{3}\gamma}=? \\ $$$$\mathrm{4}.\:\:\:\:\:\int\frac{{d}\delta}{\mathrm{cosh}\:\mathrm{2}\delta\:+\mathrm{cosh}\:\mathrm{3}\delta}=? \\ $$

Question Number 40161 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ ((sin((1/x^2 )))/(ln(1+(√x))))dx

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{sin}\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)}{{ln}\left(\mathrm{1}+\sqrt{{x}}\right)}{dx} \\ $$

Question Number 40160 Answers: 0 Comments: 1

study the convergence of ∫_0 ^1 ((1−e^(−t) )/(t(√t))) dt

$${study}\:{the}\:{convergence}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{\mathrm{1}−{e}^{−{t}} }{{t}\sqrt{{t}}}\:{dt} \\ $$

Question Number 40159 Answers: 0 Comments: 1

let I_n = ∫_0 ^∞ (dx/((1+x^3 )^n )) find a relation etween I_n and I_(n+1) 2) calculate I_(1 ) and I_2

$${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{{n}} } \\ $$$${find}\:{a}\:{relation}\:{etween}\:{I}_{{n}} \:{and}\:{I}_{{n}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}_{\mathrm{1}\:} \:{and}\:{I}_{\mathrm{2}} \\ $$

Question Number 40158 Answers: 0 Comments: 3

let A_n = ∫_0 ^1 ((x^(2n+1) ln(x))/(x^2 −1))dx 1) justify the existence of A_n 2)calculate A_(n+1) −A_n 3) prove that x∈]0,1[ ⇒0<((xln(x))/(x^2 −1))<(1/2) 4) find lim_(n→+∞) A_n

$${let}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{2}{n}+\mathrm{1}} \:{ln}\left({x}\right)}{{x}^{\mathrm{2}} \:−\mathrm{1}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{justify}\:{the}\:{existence}\:{of}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{A}_{{n}+\mathrm{1}} \:−{A}_{{n}} \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\Rightarrow\mathrm{0}<\frac{{xln}\left({x}\right)}{{x}^{\mathrm{2}} \:−\mathrm{1}}<\frac{\mathrm{1}}{\mathrm{2}}\:\:\right. \\ $$$$\left.\mathrm{4}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} {A}_{{n}} \\ $$

Question Number 40157 Answers: 1 Comments: 1

find the value of ∫_(−∞) ^(+∞) (dt/((t^2 −2t +2)^(3/2) ))

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:−\mathrm{2}{t}\:+\mathrm{2}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 40156 Answers: 1 Comments: 1

find ∫_e^2 ^(+∞) (dt/(tln(t)ln(ln(t)))

$${find}\:\:\:\int_{{e}^{\mathrm{2}} } ^{+\infty} \:\:\:\:\frac{{dt}}{{tln}\left({t}\right){ln}\left({ln}\left({t}\right)\right.} \\ $$

Question Number 40155 Answers: 1 Comments: 1

caoculate ∫_0 ^∞ ((t dt)/((1+t^4 )^2 ))

$${caoculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{t}\:{dt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$

Question Number 40154 Answers: 1 Comments: 1

find the value of ∫_0 ^1 ((ln(t))/((1+t)^2 ))dt

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$

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