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IntegrationQuestion and Answers: Page 243

Question Number 59190 Answers: 0 Comments: 0

find ∫_0 ^1 (x^2 /(1−cosx))dx .

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

Question Number 59185 Answers: 0 Comments: 1

calculate ∫_0 ^1 (x/(sinx))dx let f(x) =sinx ⇒f(x) =f(0) +xf^′ (0) +(x^2 /2)f^((2)) (0)+(x^3 /(3!))f^((3)) (0) +o(x^4 ) but f(0) =0 f^′ (x) =cosx ⇒f^′ (0)=1 f^((2)) (x) =−sinx ⇒f^((2)) (0)=0 f^((3)) (0) =−cos(x) ⇒f^((3)) (0) =−1 ⇒sinx =x−(x^3 /6) +o(x^4 ) ⇒ x−(x^3 /3) ≤sinx ≤x for x ∈]0,1] (1/x) ≤(1/(sinx)) ≤(1/(x−(x^3 /6))) ⇒1≤(x/(sinx)) ≤(1/(1−(x^2 /6))) ⇒ 1 ≤ ∫_0 ^1 (x/(sinx)) dx ≤ ∫_0 ^1 (dx/(1−(x^2 /6))) ∫_0 ^1 (dx/(1−(x^2 /6))) =_(x =(√6)t) ∫_0 ^(1/(√6)) (((√6)dt)/(1−t^2 )) =((√6)/2) ∫_0 ^(1/(√6)) ((1/(1−t)) +(1/(1+t)))dt =((√6)/2)[ln∣((1+t)/(1−t))∣]_0 ^(1/(√6)) =((√6)/2) { ln(((1+(1/(√6)))/(1−(1/(√6)))))} =((√6)/2) ln((((√6)+1)/((√6)−1))) ⇒ 1≤ ∫_0 ^1 (x/(sinx)) dx ≤ ((√6)/2)ln((((√6)+1)/((√6)−1))) .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}}{{sinx}}{dx} \\ $$$${let}\:{f}\left({x}\right)\:={sinx}\:\Rightarrow{f}\left({x}\right)\:={f}\left(\mathrm{0}\right)\:+{xf}^{'} \left(\mathrm{0}\right)\:+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}{f}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)+\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}{f}^{\left(\mathrm{3}\right)} \left(\mathrm{0}\right)\:+{o}\left({x}^{\mathrm{4}} \right) \\ $$$${but}\:{f}\left(\mathrm{0}\right)\:=\mathrm{0}\:\:\:{f}^{'} \left({x}\right)\:={cosx}\:\Rightarrow{f}^{'} \left(\mathrm{0}\right)=\mathrm{1}\:\:\:\:{f}^{\left(\mathrm{2}\right)} \left({x}\right)\:=−{sinx}\:\Rightarrow{f}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}^{\left(\mathrm{3}\right)} \left(\mathrm{0}\right)\:=−{cos}\left({x}\right)\:\Rightarrow{f}^{\left(\mathrm{3}\right)} \left(\mathrm{0}\right)\:=−\mathrm{1}\:\Rightarrow{sinx}\:={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\:+{o}\left({x}^{\mathrm{4}} \right)\:\Rightarrow \\ $$$$\left.{x}\left.−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\:\leqslant{sinx}\:\leqslant{x}\:\:\:\:{for}\:{x}\:\in\right]\mathrm{0},\mathrm{1}\right]\:\:\:\frac{\mathrm{1}}{{x}}\:\leqslant\frac{\mathrm{1}}{{sinx}}\:\leqslant\frac{\mathrm{1}}{{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}}\:\Rightarrow\mathrm{1}\leqslant\frac{{x}}{{sinx}}\:\leqslant\frac{\mathrm{1}}{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{6}}}\:\Rightarrow \\ $$$$\mathrm{1}\:\leqslant\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{x}}{{sinx}}\:{dx}\:\leqslant\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{6}}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{6}}}\:=_{{x}\:=\sqrt{\mathrm{6}}{t}} \:\:\:\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}} \:\:\:\:\frac{\sqrt{\mathrm{6}}{dt}}{\mathrm{1}−{t}^{\mathrm{2}} }\:=\frac{\sqrt{\mathrm{6}}}{\mathrm{2}}\:\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}} \:\:\left(\frac{\mathrm{1}}{\mathrm{1}−{t}}\:+\frac{\mathrm{1}}{\mathrm{1}+{t}}\right){dt} \\ $$$$=\frac{\sqrt{\mathrm{6}}}{\mathrm{2}}\left[{ln}\mid\frac{\mathrm{1}+{t}}{\mathrm{1}−{t}}\mid\right]_{\mathrm{0}} ^{\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}} \:\:=\frac{\sqrt{\mathrm{6}}}{\mathrm{2}}\:\left\{\:{ln}\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}}{\mathrm{1}−\frac{\mathrm{1}}{\sqrt{\mathrm{6}}}}\right)\right\}\:=\frac{\sqrt{\mathrm{6}}}{\mathrm{2}}\:{ln}\left(\frac{\sqrt{\mathrm{6}}+\mathrm{1}}{\sqrt{\mathrm{6}}−\mathrm{1}}\right)\:\Rightarrow \\ $$$$\mathrm{1}\leqslant\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}}{{sinx}}\:{dx}\:\leqslant\:\frac{\sqrt{\mathrm{6}}}{\mathrm{2}}{ln}\left(\frac{\sqrt{\mathrm{6}}+\mathrm{1}}{\sqrt{\mathrm{6}}−\mathrm{1}}\right)\:. \\ $$$$ \\ $$

Question Number 59184 Answers: 0 Comments: 0

find ∫ ((cos(3x))/(ch(2x)))dx .

$${find}\:\:\:\int\:\:\:\:\frac{{cos}\left(\mathrm{3}{x}\right)}{{ch}\left(\mathrm{2}{x}\right)}{dx}\:. \\ $$

Question Number 59183 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(1+ix))/(2+x^2 )) dx

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

Question Number 59175 Answers: 0 Comments: 1

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

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

Question Number 59174 Answers: 0 Comments: 0

calculate ∫∫_([0,2]^2 ) (x+1−(√y))(y+1−(√x))dxdy

$${calculate}\:\int\int_{\left[\mathrm{0},\mathrm{2}\right]^{\mathrm{2}} } \:\:\:\:\left({x}+\mathrm{1}−\sqrt{{y}}\right)\left({y}+\mathrm{1}−\sqrt{{x}}\right){dxdy}\: \\ $$

Question Number 59172 Answers: 0 Comments: 1

calculate ∫∫_([1,3]^2 ) (x+y)ln(x^2 +y^2 )dxdy

$${calculate}\:\int\int_{\left[\mathrm{1},\mathrm{3}\right]^{\mathrm{2}} } \:\:\:\:\left({x}+{y}\right){ln}\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){dxdy}\: \\ $$

Question Number 59169 Answers: 0 Comments: 1

calculate A_n =∫∫_([(1/n),n[^2 ) e^(−x^2 −3y^2 ) (√(x^2 +3y^2 ))dxdy and find lim_(n→+∞) A_n

$${calculate}\:{A}_{{n}} =\int\int_{\left[\frac{\mathrm{1}}{{n}},{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\:{e}^{−{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} } \sqrt{{x}^{\mathrm{2}} \:+\mathrm{3}{y}^{\mathrm{2}} }{dxdy} \\ $$$${and}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 59168 Answers: 1 Comments: 0

calculate ∫_(−1) ^1 ((x^2 +3)/((√(1+x)) +(√(1−x)))) dx

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

Question Number 59167 Answers: 0 Comments: 0

find the value of Σ_(n=1) ^∞ ((n^2 +1)/(n^3 (n+1)^2 ))

$${find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}^{\mathrm{2}} \:+\mathrm{1}}{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 59165 Answers: 0 Comments: 0

let D ={ (x,y)∈R^2 / x>0 ,y>0 and x+y ≤2 } calculate ∫∫_D (x+y −(√(x+y)))dxdy

$${let}\:{D}\:=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:{x}>\mathrm{0}\:,{y}>\mathrm{0}\:\:{and}\:\:{x}+{y}\:\leqslant\mathrm{2}\:\right\} \\ $$$${calculate}\:\int\int_{{D}} \:\left({x}+{y}\:−\sqrt{{x}+{y}}\right){dxdy}\: \\ $$

Question Number 59163 Answers: 0 Comments: 0

calculate A_n =∫∫_W_n ((1−(√(x^2 +y^2 )))/(1+(√(x^2 +y^2 )))) dxdy with W_n =](1/n),n[^2 2) find lim_(n→+∞) A_n

$$\left.{calculate}\:{A}_{{n}} =\int\int_{{W}_{{n}} } \:\:\:\frac{\mathrm{1}−\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}{\mathrm{1}+\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}\:{dxdy}\:\:\:{with}\:{W}_{{n}} \:=\right]\frac{\mathrm{1}}{{n}},{n}\left[^{\mathrm{2}} \right. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 59162 Answers: 0 Comments: 0

calculate ∫∫_D (√(x^2 −y^2 ))xy dxdy with D ={(x,y)∈ R^2 /0≤y≤1 and 2 ≤x ≤5 }

$${calculate}\:\:\int\int_{{D}} \:\:\sqrt{{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }{xy}\:{dxdy}\:\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in\:{R}^{\mathrm{2}} \:\:\:/\mathrm{0}\leqslant{y}\leqslant\mathrm{1}\:\:{and}\:\:\mathrm{2}\:\leqslant{x}\:\leqslant\mathrm{5}\:\right\} \\ $$

Question Number 59161 Answers: 0 Comments: 0

calculatef(a)= ∫_0 ^∞ ((ln(a^2 +x^2 ))/(a^2 +x^2 ))dx with >0 1) calculate ∫_0 ^∞ ((ln(2+x^2 ))/(2+x^2 ))dx and ∫_0 ^∞ ((ln(3+x^2 ))/(3+x^2 )) dx .

$${calculatef}\left({a}\right)=\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left({a}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)}{{a}^{\mathrm{2}} +{x}^{\mathrm{2}} }{dx}\:\:\:{with}\:>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)}{\mathrm{2}+{x}^{\mathrm{2}} }{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{3}+{x}^{\mathrm{2}} \right)}{\mathrm{3}+{x}^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 59052 Answers: 0 Comments: 0

let f(x) =∫_0 ^(π/2) ln(cosθ +xsinθ)dθ with x fromR 1) determine a explicit form for f(x) 2) calculate ∫_0 ^(π/2) ln(cosθ +sinθ) dθ and ∫_0 ^(π/2) ln(cosθ +2sinθ)dθ .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cos}\theta\:+{xsin}\theta\right){d}\theta\:\:\:{with}\:{x}\:{fromR} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cos}\theta\:+{sin}\theta\right)\:{d}\theta\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cos}\theta\:+\mathrm{2}{sin}\theta\right){d}\theta\:. \\ $$

Question Number 59050 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 )) dx

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

Question Number 59040 Answers: 3 Comments: 0

1) ∫_0 ^( (π/4)) ((sin x+cos x)/(cos^2 x+sin^4 x))dx = ?

$$\left.\mathrm{1}\right)\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{cos}\:^{\mathrm{2}} {x}+\mathrm{sin}\:^{\mathrm{4}} {x}}{dx}\:=\:? \\ $$

Question Number 59012 Answers: 3 Comments: 0

Question Number 58940 Answers: 1 Comments: 3

e^(i∫_(−2) ^2 (x^2 sinx+(√(1−(x^2 /4))))dx) +lim_(x→2) ((∫_2 ^x log(x+8)dx)/(x−2))=?

$${e}^{{i}\int_{−\mathrm{2}} ^{\mathrm{2}} \left({x}^{\mathrm{2}} {sinx}+\sqrt{\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}}\right){dx}} +\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\int_{\mathrm{2}} ^{{x}} {log}\left({x}+\mathrm{8}\right){dx}}{{x}−\mathrm{2}}=? \\ $$

Question Number 58937 Answers: 0 Comments: 2

∫_0 ^2 lim_((1/n)→0) (((2−x)(x+x^n ))/(1+x^n ))dx= ?

$$\int_{\mathrm{0}} ^{\mathrm{2}} \underset{\frac{\mathrm{1}}{{n}}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{2}−{x}\right)\left({x}+{x}^{{n}} \right)}{\mathrm{1}+{x}^{{n}} }{dx}=\:? \\ $$

Question Number 58791 Answers: 1 Comments: 0

Show that: ∫_( 0) ^( ∞) ((sin(x))/x) = (π/2)

$$\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}:\:\:\:\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\infty} \:\:\:\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{x}}}\:\:\:=\:\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 58774 Answers: 0 Comments: 3

let f(x) =∫_(π/3) ^(π/2) (dθ/(1+xtanθ)) with x real 1) find a explicit form for f(x) 2) determine also g(x) =∫_(π/3) ^(π/2) ((tanθ)/((1+xtanθ)^2 )) dθ 3) let U_n (x) =f^((n)) (x) give U_n (x) at form of integral. 4) calculate ∫_(π/3) ^(π/2) (dθ/(1+2tanθ)) and ∫_(π/3) ^(π/2) ((tanθ dθ)/((1+2tanθ)^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{d}\theta}{\mathrm{1}+{xtan}\theta}\:\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({x}\right)\:=\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{tan}\theta}{\left(\mathrm{1}+{xtan}\theta\right)^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{let}\:{U}_{{n}} \left({x}\right)\:={f}^{\left({n}\right)} \left({x}\right)\:\:{give}\:{U}_{{n}} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{d}\theta}{\mathrm{1}+\mathrm{2}{tan}\theta}\:\:{and}\:\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{tan}\theta\:{d}\theta}{\left(\mathrm{1}+\mathrm{2}{tan}\theta\right)^{\mathrm{2}} } \\ $$

Question Number 58770 Answers: 2 Comments: 1

find the value of integrals I =∫_0 ^∞ (dx/((x^2 +1)^3 )) , J =∫_0 ^∞ (dx/((x^2 +1)^5 ))

$${find}\:{the}\:{value}\:{of}\:{integrals} \\ $$$$\:{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }\:\:\:,\:{J}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{5}} } \\ $$

Question Number 58717 Answers: 1 Comments: 1

Question Number 58648 Answers: 2 Comments: 4

Question Number 58639 Answers: 1 Comments: 0

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