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Question Number 173005    Answers: 0   Comments: 1

Question Number 173004    Answers: 0   Comments: 1

Question Number 173003    Answers: 0   Comments: 0

Question Number 173002    Answers: 1   Comments: 1

Question Number 173001    Answers: 2   Comments: 0

Question Number 173000    Answers: 0   Comments: 1

Question Number 172999    Answers: 1   Comments: 0

Question Number 172996    Answers: 1   Comments: 0

let U_n =∫_0 ^1 (√(1−x^n ))ln^2 xdx 1)lim U_n ? 2)equivalent of U_n (n→∞)

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{{n}} }{ln}^{\mathrm{2}} {xdx} \\ $$$$\left.\mathrm{1}\right){lim}\:{U}_{{n}} ? \\ $$$$\left.\mathrm{2}\right){equivalent}\:{of}\:{U}_{{n}} \left({n}\rightarrow\infty\right) \\ $$

Question Number 181403    Answers: 1   Comments: 0

Refer to Q181319 ∫((x^4 +1)/(x^4 +x+2))dx ((x^4 +1)/(x^4 +x+2))=1+(x/(2(x^2 +x+1)))−(x/(2(x^2 −x+1))) =1+(1/4)[(((2x+1)−1)/(x^2 +x+1))−(1/4)×(((2x−1)+1)/((x^2 −x+1))) =(1/4)×[log(x^2 +x+1)]^′ −(1/4)×((1/([((√3)/2)(x+(1/2))^2 ]+1))) −((1/4)[log(x^2 −x+1)]^′ +(1/4)((1/([((√3)/2)×(x−(1/2)]^2 +1)))) first with [((√3)/2)( x+(1/2))]=u ∫(1/(u^2 +1))du=arctg(u) and v=[((√3)/2)(x−(1/2) )]=v ∫(dv/(1+v^2 ))=arctg(v) ...............

$${Refer}\:{to}\:\mathrm{Q181319} \\ $$$$\int\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{4}} +{x}+\mathrm{2}}{dx} \\ $$$$\frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{4}} +{x}+\mathrm{2}}=\mathrm{1}+\frac{{x}}{\mathrm{2}\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)}−\frac{{x}}{\mathrm{2}\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)} \\ $$$$=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}\left[\frac{\left(\mathrm{2}{x}+\mathrm{1}\right)−\mathrm{1}}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{4}}×\frac{\left(\mathrm{2}{x}−\mathrm{1}\right)+\mathrm{1}}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)}\right. \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}×\left[\mathrm{log}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)\right]^{'} −\frac{\mathrm{1}}{\mathrm{4}}×\left(\frac{\mathrm{1}}{\left[\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \right]+\mathrm{1}}\right) \\ $$$$−\left(\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{log}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)\right]^{'} +\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{1}}{\left[\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}×\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right]^{\mathrm{2}} +\mathrm{1}\right.}\right)\right) \\ $$$${first}\:{with}\:\left[\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\left(\:{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)\right]={u}\:\:\:\int\frac{\mathrm{1}}{{u}^{\mathrm{2}} +\mathrm{1}}{du}={arctg}\left({u}\right) \\ $$$${and}\:{v}=\left[\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\:\right)\right]=\mathrm{v}\:\:\:\int\frac{{dv}}{\mathrm{1}+{v}^{\mathrm{2}} }={arctg}\left({v}\right) \\ $$$$............... \\ $$

Question Number 172991    Answers: 2   Comments: 0

Question Number 172989    Answers: 1   Comments: 0

calcul: R=+oo Σ_(n=o) ^(+oo) ((2n)/((2n+1)!))x^n

$${calcul}:\:{R}=+{oo}\: \\ $$$$\underset{{n}={o}} {\overset{+{oo}} {\sum}}\frac{\mathrm{2}{n}}{\left(\mathrm{2}{n}+\mathrm{1}\right)!}{x}^{{n}} \\ $$

Question Number 172984    Answers: 1   Comments: 0

Question Number 181400    Answers: 0   Comments: 2

Question Number 172973    Answers: 1   Comments: 0

∫_0 ^( ∞) (( 1)/((1+ x^( 4) ) (1+ x^( 6) )))dx=((p(√2) −q)/(12)) π p , q=?

$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{1}}{\left(\mathrm{1}+\:{x}^{\:\mathrm{4}} \right)\:\left(\mathrm{1}+\:{x}^{\:\mathrm{6}} \right)}{dx}=\frac{{p}\sqrt{\mathrm{2}}\:−{q}}{\mathrm{12}}\:\pi \\ $$$$\:\:\:{p}\:,\:\:{q}=? \\ $$$$ \\ $$

Question Number 181394    Answers: 1   Comments: 0

Question Number 172956    Answers: 0   Comments: 2

Question Number 172953    Answers: 2   Comments: 0

find ∫_0 ^1 (√(1−x^4 ))lnx dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{lnx}\:{dx} \\ $$

Question Number 172949    Answers: 0   Comments: 0

Question Number 172948    Answers: 1   Comments: 6

Question Number 172947    Answers: 1   Comments: 0

Question Number 172944    Answers: 1   Comments: 1

∫∫_d dxdy x=y^2 −1 x=1−y D=?

$$\int\int_{{d}} {dxdy}\:\:\:\:\:\:{x}={y}^{\mathrm{2}} −\mathrm{1}\:\:\:\:\:\:{x}=\mathrm{1}−{y} \\ $$$${D}=? \\ $$

Question Number 172929    Answers: 0   Comments: 1

A man purchased 180 copies of a book at $250.00 each. He sold y copies at $300.00 each and the rest at a discount of 5 cents in the dollar of the cost price. If he made a profit of $7,125.00,find the value of y?

$$\mathrm{A}\:\mathrm{man}\:\mathrm{purchased}\:\mathrm{180}\:\mathrm{copies}\:\mathrm{of}\:\mathrm{a}\:\mathrm{book} \\ $$$$\mathrm{at}\:\$\mathrm{250}.\mathrm{00}\:\boldsymbol{\mathrm{each}}.\:\mathrm{He}\:\mathrm{sold}\:{y}\:\mathrm{copies}\:\mathrm{at}\:\$\mathrm{300}.\mathrm{00} \\ $$$$\mathrm{each}\:\mathrm{and}\:\mathrm{the}\:\mathrm{rest}\:\mathrm{at}\:\mathrm{a}\:\mathrm{discount}\:\mathrm{of}\:\mathrm{5}\:\mathrm{cents}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{dollar}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cost}\:\mathrm{price}.\:\mathrm{If}\:\mathrm{he}\:\mathrm{made}\:\mathrm{a}\:\mathrm{profit} \\ $$$$\mathrm{of}\:\$\mathrm{7},\mathrm{125}.\mathrm{00},\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{y}? \\ $$

Question Number 172923    Answers: 0   Comments: 0

Question Number 172918    Answers: 0   Comments: 9

Question Number 172915    Answers: 3   Comments: 1

Question Number 172913    Answers: 2   Comments: 0

lim_(x→∞) ((√(x^2 +2x+3))−(√(x^2 +3)) )^x =?

$$\:\:\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{3}}−\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}\:\right)^{{x}} \:=? \\ $$

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