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Question Number 166373 Answers: 0 Comments: 0

prove 𝛗=∫_0 ^( 1) (( ln^( 2) (1−x^( 2) ) )/x^( 2) ) dx =(π^( 2) /3) −4ln^( 2) (2) −−− solution (technical method) −−− 𝛗= ∫_0 ^( 1) ln^( 2) (1−x^( 2) )d(1−(1/x)) = [(1−(1/x))ln^( 2) (1−x^( 2) )]_0 ^1 +4∫_0 ^( 1) (1−(1/x))((xln(1−x^( 2) ))/(1−x^( 2) ))dx = −4∫_0 ^( 1) ((ln(1−x^( 2) ))/(1+x)) dx = −4∫_0 ^( 1) ((ln(1+x))/(1+x))dx −4∫_0 ^( 1) ((ln(1−x)dx)/(1+x)) = −2ln^( 2) (2) −4 ( −(π^( 2) /(12)) +(1/2)ln^( 2) (2)) ∴ 𝛗= (π^( 2) /3) −4ln^( 2) (2) ■ m.n

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{prove}\:\: \\ $$$$\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{\:\mathrm{2}} \right)\:}{{x}^{\:\mathrm{2}} }\:{dx}\:=\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:−\mathrm{4}{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:−−−\:\:{solution}\:\left({technical}\:{method}\right)\:−−− \\ $$$$\:\:\:\:\boldsymbol{\phi}=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{\:\mathrm{2}} \right){d}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right) \\ $$$$\:\:\:\:\:\:\:\:=\:\left[\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right){ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{\:\mathrm{2}} \right)\right]_{\mathrm{0}} ^{\mathrm{1}} +\mathrm{4}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)\frac{{xln}\left(\mathrm{1}−{x}^{\:\mathrm{2}} \right)}{\mathrm{1}−{x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:=\:−\mathrm{4}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}^{\:\mathrm{2}} \right)}{\mathrm{1}+{x}}\:{dx}\: \\ $$$$\:\:\:\:\:\:\:=\:−\mathrm{4}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}}{dx}\:−\mathrm{4}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right){dx}}{\mathrm{1}+{x}} \\ $$$$\:\:\:\:\:\:=\:−\mathrm{2}{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\:−\mathrm{4}\:\left(\:−\frac{\pi^{\:\mathrm{2}} }{\mathrm{12}}\:+\frac{\mathrm{1}}{\mathrm{2}}{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\therefore\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{3}}\:−\mathrm{4}{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 166371 Answers: 0 Comments: 2

Question Number 166372 Answers: 3 Comments: 0

Question Number 166360 Answers: 1 Comments: 0

((sin 10x)/(sin 2x))−((cos 10x)/(cos 2x))=?

$$\frac{\mathrm{sin}\:\mathrm{10}{x}}{{sin}\:\mathrm{2}{x}}−\frac{\mathrm{cos}\:\mathrm{10}{x}}{\mathrm{cos}\:\mathrm{2}{x}}=? \\ $$

Question Number 166358 Answers: 1 Comments: 0

Question Number 166350 Answers: 1 Comments: 1

Question Number 166331 Answers: 2 Comments: 1

prove that ((df^(−1) (a))/dx)×((df(f^(−1) (a)))/dx)=1

$${prove}\:{that} \\ $$$$\frac{{df}^{−\mathrm{1}} \left({a}\right)}{{dx}}×\frac{{df}\left({f}^{−\mathrm{1}} \left({a}\right)\right)}{{dx}}=\mathrm{1} \\ $$

Question Number 166321 Answers: 1 Comments: 0

Question Number 166320 Answers: 1 Comments: 0

∫ (dx/(tan^2 x+sin^2 x)) =?

$$\:\:\int\:\frac{\mathrm{dx}}{\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}\:=? \\ $$

Question Number 166319 Answers: 1 Comments: 2

Question Number 166318 Answers: 0 Comments: 0

Question Number 166346 Answers: 1 Comments: 1

∫(dx/(1+(√x)+(√(1+x))))

$$\int\frac{{dx}}{\mathrm{1}+\sqrt{{x}}+\sqrt{\mathrm{1}+{x}}} \\ $$

Question Number 166307 Answers: 1 Comments: 0

∫_0 ^1 ((x^4 (1−x)^4 )/(1+x^2 ))dx

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

Question Number 166296 Answers: 0 Comments: 2

(cos x)^(2022) −(sin x)^(2022) =1 x=?

$$\left(\mathrm{cos}\:{x}\right)^{\mathrm{2022}} −\left(\mathrm{sin}\:{x}\right)^{\mathrm{2022}} =\mathrm{1} \\ $$$${x}=? \\ $$

Question Number 166294 Answers: 0 Comments: 5

f(x)=∫_1 ^x (dt/( (√(t^3 +2t^2 +3)))) (f^(−1) (0))′=?

$$\:\mathrm{f}\left(\mathrm{x}\right)=\int_{\mathrm{1}} ^{{x}} \:\frac{{dt}}{\:\sqrt{{t}^{\mathrm{3}} +\mathrm{2}{t}^{\mathrm{2}} +\mathrm{3}}} \\ $$$$\:\:\:\left({f}^{−\mathrm{1}} \left(\mathrm{0}\right)\right)'=? \\ $$

Question Number 166291 Answers: 1 Comments: 0

calculer la somme Σ_(n=1) ^(+oo) (1/(n(n+2)))x^n

$${calculer}\:{la}\:{somme} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{+{oo}} {\sum}}\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{2}\right)}{x}^{{n}} \\ $$

Question Number 166284 Answers: 0 Comments: 1

Question Number 166281 Answers: 1 Comments: 0

Question Number 166280 Answers: 1 Comments: 0

Solve for the exact value of ∫_0 ^( ∞) sin(x^2 + x^(−2) ) dx

$$\:\mathrm{Solve}\:\mathrm{for}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\:\infty} \mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}^{−\mathrm{2}} \right)\:\mathrm{dx} \\ $$

Question Number 166279 Answers: 0 Comments: 0

if y = x^x^x^x find y^′ ?

$${if}\:{y}\:=\:{x}^{{x}^{{x}^{{x}} } } \:{find}\:{y}^{'} \:? \\ $$

Question Number 166273 Answers: 2 Comments: 1

Question Number 166263 Answers: 1 Comments: 0

∫_( −(π/4)) ^( (π/4)) (dx/(cos^2 x (√(9+7 tan ∣x∣)))) dx =?

$$\:\:\:\:\:\:\int_{\:−\frac{\pi}{\mathrm{4}}} ^{\:\frac{\pi}{\mathrm{4}}} \frac{\mathrm{dx}}{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\sqrt{\mathrm{9}+\mathrm{7}\:\mathrm{tan}\:\mid\mathrm{x}\mid}}\:\mathrm{dx}\:=? \\ $$

Question Number 166301 Answers: 2 Comments: 0

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

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

Question Number 166261 Answers: 1 Comments: 1

If four men can dig a piece of land in 2days at eight hours everyday.How many days can two men dig in every six hours perday?

$${If}\:{four}\:{men}\:{can}\:{dig}\:{a}\:{piece}\:{of}\:{land}\:{in}\:\mathrm{2}{days}\:{at}\:{eight}\:{hours}\:{everyday}.{How}\:{many}\:{days}\:{can}\:{two}\:{men}\:{dig}\:{in}\:{every}\:{six}\:{hours}\:{perday}? \\ $$

Question Number 166254 Answers: 1 Comments: 0

Θ=Σ_(n=1) ^∞ (( H_( n) )/(n. (n+1 ))) =^? (π^( 2) /6) −−−−+

$$ \\ $$$$\:\:\:\:\Theta=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{H}_{\:{n}} }{{n}.\:\left({n}+\mathrm{1}\:\right)}\:\:\overset{?} {=}\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{6}} \\ $$$$\:\:\:\:\:−−−−+ \\ $$

Question Number 166252 Answers: 1 Comments: 1

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