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Let I_1 = ∫_(π/6) ^(π/3) ((sin x)/x) dx , I_2 = ∫_(π/6) ^(π/3) ((sin (sin x))/(sin x))dx , I_3 = ∫_(π/6) ^(π/3) ((sin (tan x))/(tan x))dx. Prove that I_2 > I_1 > I_3 .

$$\mathrm{Let}\:\mathrm{I}_{\mathrm{1}} =\:\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{sin}\:{x}}{{x}}\:{dx}\:\:,\:\:\mathrm{I}_{\mathrm{2}} =\:\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)}{\mathrm{sin}\:{x}}{dx} \\ $$$$,\:\mathrm{I}_{\mathrm{3}} =\:\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} \frac{\mathrm{sin}\:\left(\mathrm{tan}\:{x}\right)}{\mathrm{tan}\:{x}}{dx}.\: \\ $$$${P}\mathrm{rove}\:\mathrm{that}\:\mathrm{I}_{\mathrm{2}} \:>\:\mathrm{I}_{\mathrm{1}} \:>\:\mathrm{I}_{\mathrm{3}} \:. \\ $$

Question Number 40760 Answers: 0 Comments: 0

find ∫ ((√(1+x^2 ))/(√(1−x^3 ))) dx

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

Question Number 40745 Answers: 3 Comments: 0

Question Number 40717 Answers: 1 Comments: 1

∫(√(tanx/sinx.cosxdx))

$$\int\sqrt{{tanx}/{sinx}.{cosxdx}} \\ $$

Question Number 40716 Answers: 2 Comments: 0

∫(cosx−cos2x/1−cosx)dx

$$\int\left({cosx}−{cos}\mathrm{2}{x}/\mathrm{1}−{cosx}\right){dx} \\ $$

Question Number 40684 Answers: 0 Comments: 1

∫((x^7 −1)/(logx))dx

$$\int\frac{\mathrm{x}^{\mathrm{7}} −\mathrm{1}}{\mathrm{logx}}\mathrm{dx} \\ $$

Question Number 40675 Answers: 1 Comments: 2

Question Number 40661 Answers: 0 Comments: 4

1)find g(x)=∫_0 ^(π/2) ln(1−x^2 cos^2 θ)dθ with x from R 2) find the value of ∫_0 ^(π/2) ln(1−2 cos^2 θ)dθ and 3) find the value of A(α)=∫_0 ^(π/2) ln(1−cos^2 α cos^2 θ)dθ

$$\left.\mathrm{1}\right){find}\:\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−{x}^{\mathrm{2}} {cos}^{\mathrm{2}} \theta\right){d}\theta\:\:{with}\:{x}\:{from}\:{R} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−\mathrm{2}\:{cos}^{\mathrm{2}} \theta\right){d}\theta\:{and} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\: \\ $$$${A}\left(\alpha\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}−{cos}^{\mathrm{2}} \alpha\:{cos}^{\mathrm{2}} \theta\right){d}\theta\: \\ $$

Question Number 40660 Answers: 0 Comments: 0

let f(t) = ∫_0 ^∞ ((arctan(tcosx))/(1+x^2 ))dx 1) find another form of f(t) 2) calculate ∫_0 ^∞ ((arctan(2cosx))/(1+x^2 ))dx .

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({tcosx}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{another}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{arctan}\left(\mathrm{2}{cosx}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 40658 Answers: 0 Comments: 1

find ∫_0 ^(π/4) ((x−1)/(2+cosx))dx .

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{x}−\mathrm{1}}{\mathrm{2}+{cosx}}{dx}\:. \\ $$

Question Number 40657 Answers: 1 Comments: 0

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.

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

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