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IntegrationQuestion and Answers: Page 244

Question Number 58639 Answers: 1 Comments: 0

Question Number 58627 Answers: 1 Comments: 3

Question Number 58626 Answers: 1 Comments: 1

Question Number 58595 Answers: 1 Comments: 0

Question Number 58488 Answers: 0 Comments: 2

let f(x) =∫ (dt/(x +cost +cos(2t))) (x real) 1) find a explicit form of f(x) 2)determine also ∫ (dt/((x+cost +cos(2t))^2 )) 3) find ∫ (dt/(1+cos(t)+cos(2t))) and ∫ (dt/((3 +cos(t)+cos(2t))^2 ))

$${let}\:{f}\left({x}\right)\:=\int\:\:\:\frac{{dt}}{{x}\:+{cost}\:+{cos}\left(\mathrm{2}{t}\right)}\:\:\left({x}\:{real}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){determine}\:{also}\:\int\:\:\frac{{dt}}{\left({x}+{cost}\:+{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:\int\:\:\:\frac{{dt}}{\mathrm{1}+{cos}\left({t}\right)+{cos}\left(\mathrm{2}{t}\right)}\:{and} \\ $$$$\int\:\:\:\frac{{dt}}{\left(\mathrm{3}\:+{cos}\left({t}\right)+{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} } \\ $$

Question Number 58487 Answers: 0 Comments: 3

let f(x) =∫_(π/4) ^(π/3) (dt/(2+xsint)) 1) find a explicit form of f(x) 2)determine also g(x)=∫_(π/4) ^(π/3) ((sint)/((2+xsint)^2 ))dt 3) find the value of ∫_(π/4) ^(π/3) (dt/(2+3sint)) and ∫_(π/4) ^(π/3) ((sint)/((2+3sint)^2 ))dt

$${let}\:{f}\left({x}\right)\:=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{dt}}{\mathrm{2}+{xsint}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){determine}\:{also}\:{g}\left({x}\right)=\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{sint}}{\left(\mathrm{2}+{xsint}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\frac{{dt}}{\mathrm{2}+\mathrm{3}{sint}} \\ $$$${and}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{sint}}{\left(\mathrm{2}+\mathrm{3}{sint}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 58478 Answers: 2 Comments: 1

{1} ∫((x^2 −2)/(x^4 +8x^2 +4)) dx = ? {2} Shortest distance between the parabolas y^2 =4x and y^2 =2x−6 is ?

$$\left\{\mathrm{1}\right\}\:\:\:\int\frac{{x}^{\mathrm{2}} −\mathrm{2}}{{x}^{\mathrm{4}} +\mathrm{8}{x}^{\mathrm{2}} +\mathrm{4}}\:{dx}\:=\:? \\ $$$$\left\{\mathrm{2}\right\}\:\:{Shortest}\:{distance}\:{between}\:{the} \\ $$$${parabolas}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:{and}\:{y}^{\mathrm{2}} =\mathrm{2}{x}−\mathrm{6}\:{is}\:? \\ $$

Question Number 58319 Answers: 1 Comments: 0

∫x^n (lnx)^n dx

$$\int{x}^{{n}} \left(\mathrm{ln}{x}\right)^{{n}} {dx} \\ $$

Question Number 58299 Answers: 1 Comments: 1

find ∫ (dx/((x^2 +x+1)^(3/2) ))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{x}+\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 58259 Answers: 4 Comments: 4

a .∫ (dx/(2sin^2 x+3tg^2 x))=? b .∫(( 1+(x)^(1/3) )/(1+(√x)+(x)^(1/3) +(x)^(1/6) ))dx=? c .∫ ((cosx)/(1+cos2x))dx=? d .∫ ((sin^2 x)/((√2)+(√3).cos^2 x))dx=?

Question Number 58250 Answers: 1 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3x}^{\mathrm{3}} \:−\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{2x}\:−\:\mathrm{4}}{\sqrt{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3x}\:+\:\mathrm{2}}}\right)\:\mathrm{dx} \\ $$

Question Number 58249 Answers: 0 Comments: 0

I_n ^ =∫_0 ^(π/2) cos^n xcos(nx)dx then show that I_1 ,I_2 ,I_3 ....are in G.P

$$\overset{} {{I}}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {cos}^{{n}} {xcos}\left({nx}\right){dx} \\ $$$${then}\:{show}\:{that}\:{I}_{\mathrm{1}} ,{I}_{\mathrm{2}} ,{I}_{\mathrm{3}} ....{are}\:{in}\:{G}.{P} \\ $$

Question Number 58240 Answers: 1 Comments: 2

i=∫dx/(ax^2 +bx+c)^(3/2)

$$\mathrm{i}=\int\mathrm{dx}/\left(\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$

Question Number 58238 Answers: 0 Comments: 0

∫_( 0) ^( 1) (((3x^3 − x^2 + 2x − 4)/(√(x^2 − 3x + 2)))) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\:\left(\frac{\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{3}} \:−\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:+\:\mathrm{2}\boldsymbol{\mathrm{x}}\:−\:\mathrm{4}}{\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} \:−\:\mathrm{3}\boldsymbol{\mathrm{x}}\:+\:\mathrm{2}}}\right)\:\:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 58222 Answers: 2 Comments: 4

∫_0 ^1 x^x dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{x}} {dx} \\ $$

Question Number 58220 Answers: 1 Comments: 0

find ∫ (dx/((x^2 +x)(√(−x^2 +2x +3))))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}\right)\sqrt{−{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:+\mathrm{3}}} \\ $$$$ \\ $$

Question Number 58212 Answers: 0 Comments: 0

let f(x) =∫_0 ^∞ e^(−x[t]) sin(xt)dt with x>0 1) find a explicit form for f(x) 2) let U_n =nf(n) find lim_(n→+∞) U_n and study the convergence of ΣU_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} \:{sin}\left({xt}\right){dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{U}_{{n}} ={nf}\left({n}\right)\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{U}_{{n}} \\ $$

Question Number 58187 Answers: 0 Comments: 0

let f(x) =∫_1 ^3 arctan(x+(x/t))dt withx>0 1) determine a explicit form of f(x) 2) give f^′ (x) at form of integral and find its value 3) calculate ∫_1 ^3 arctan(1+(1/t))dt and ∫_1 ^3 arctan(2+(2/t))dt . 4) calculate ∫_1 ^3 (2t−1)arctan(1+(1/t))dt .

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left({x}+\frac{{x}}{{t}}\right){dt}\:\:\:{withx}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{give}\:{f}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{find}\:{its}\:{value} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:\:\:{and}\:\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{2}+\frac{\mathrm{2}}{{t}}\right){dt}\:. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\left(\mathrm{2}{t}−\mathrm{1}\right){arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:. \\ $$

Question Number 58185 Answers: 0 Comments: 0

find ∫ ((xdx)/(cosx +sin(2x)))

$${find}\:\int\:\:\:\frac{{xdx}}{{cosx}\:+{sin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 58184 Answers: 0 Comments: 0

find ∫ ((xdx)/(sinx +cos(2x)))

$${find}\:\:\int\:\:\:\:\:\:\frac{{xdx}}{{sinx}\:+{cos}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 58168 Answers: 2 Comments: 0

find ∫ ((√(tanx))/(sin(2x)))dx

$${find}\:\int\:\:\:\:\frac{\sqrt{{tanx}}}{{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 57992 Answers: 1 Comments: 0

Question Number 57991 Answers: 0 Comments: 0

Question Number 57948 Answers: 0 Comments: 0

let A(ξ) =∫_ξ ^ξ^2 ((arctan(1+ξt)−(π/4))/((√(2+ξt))−(√(2−ξt)))) dt find lim_(ξ →0) A(ξ) .

$${let}\:{A}\left(\xi\right)\:=\int_{\xi} ^{\xi^{\mathrm{2}} } \:\:\:\:\frac{{arctan}\left(\mathrm{1}+\xi{t}\right)−\frac{\pi}{\mathrm{4}}}{\sqrt{\mathrm{2}+\xi{t}}−\sqrt{\mathrm{2}−\xi{t}}}\:{dt} \\ $$$${find}\:{lim}_{\xi\:\rightarrow\mathrm{0}} \:\:{A}\left(\xi\right)\:. \\ $$$$ \\ $$

Question Number 57900 Answers: 0 Comments: 1

let f(x) =∫_0 ^∞ ((cos(πxt))/((t^2 +3x^2 )^2 )) dt with x>0 1) find a explicit form for f(x) 2) find the value of ∫_0 ^∞ ((cos(πt))/((t^2 +3)^2 ))dt 3) let U_n =f(n) find nature of Σ U_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{xt}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{t}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{U}_{{n}} ={f}\left({n}\right)\:\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 57899 Answers: 0 Comments: 2

let f(x) =∫_0 ^(+∞) (dt/((t^2 +x^2 )^3 )) with x>0 1) find a explicit form off (x) 1) calculate ∫_0 ^∞ (dx/((t^2 +3)^3 )) and ∫_0 ^∞ (dt/((t^2 +4)^3 )) 2) find the value of A(θ) =∫_0 ^∞ (dt/((t^2 +sin^2 θ)^3 )) with 0<θ<π.

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{off}\:\left({x}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{3}} }\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{sin}^{\mathrm{2}} \theta\right)^{\mathrm{3}} }\:\:{with}\:\mathrm{0}<\theta<\pi. \\ $$

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