Question and Answers Forum

All Questions Topic List

IntegrationQuestion and Answers: Page 59

Question Number 162112 Answers: 1 Comments: 0

∫(( cos(x))/((1−cos(x))^2 ))dx=?

$$\int\frac{\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}{\left(\mathrm{1}−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$

Question Number 162099 Answers: 0 Comments: 0

PROVE THAT Ω= ∫_0 ^( 1) (( Li_( 2) (x ). ln( x ))/x) dx =^? (( −π^( 4) )/(90)) −−−−−−−−−− Ω= ∫_0 ^( 1) ln (x )Σ_(n=1) ^∞ (( x^( n−1) )/n^( 2) ) dx = Σ_(n=1) ^∞ (1/n^( 2) ) ∫_0 ^( 1) x^( n−1) . ln(x ) dx = Σ_(n=1) ^∞ (1/n^( 2) ) {[ (x^( n) /n) ln( x )]_0 ^( 1) −(1/n) ∫_0 ^( 1) x^( n−1) dx} = Σ_(n=1) ^∞ ((−1)/n^( 4) ) = − ζ (4 ) = ((−π^( 4) )/( 90)) ■ m.n −−− M . N −−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{PROVE}\:\:\:\mathscr{THAT}\:\: \\ $$$$\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{Li}_{\:\mathrm{2}} \:\left({x}\:\right).\:\mathrm{ln}\left(\:{x}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:−\pi^{\:\mathrm{4}} }{\mathrm{90}} \\ $$$$\:\:\:\:\:−−−−−−−−−− \\ $$$$\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{ln}\:\left({x}\:\right)\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{x}^{\:{n}−\mathrm{1}} }{{n}^{\:\mathrm{2}} }\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:{x}^{\:{n}−\mathrm{1}} .\:\mathrm{ln}\left({x}\:\right)\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\:\mathrm{2}} }\:\left\{\left[\:\frac{{x}^{\:{n}} }{{n}}\:\mathrm{ln}\left(\:{x}\:\right)\right]_{\mathrm{0}} ^{\:\mathrm{1}} −\frac{\mathrm{1}}{{n}}\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\:{n}−\mathrm{1}} {dx}\right\} \\ $$$$\:\:\:\:\:\:\:\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{−\mathrm{1}}{{n}^{\:\mathrm{4}} }\:=\:−\:\zeta\:\left(\mathrm{4}\:\right)\:=\:\frac{−\pi^{\:\mathrm{4}} }{\:\mathrm{90}}\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:−−−\:\mathscr{M}\:.\:\mathscr{N}\:\:−−−\: \\ $$$$ \\ $$

Question Number 162117 Answers: 2 Comments: 1

Question Number 162066 Answers: 0 Comments: 0

Σ_(n=1) ^∞ (((−1)^n H_n )/n^2 )=???

$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{H}}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}^{\mathrm{2}} }=??? \\ $$

Question Number 162055 Answers: 1 Comments: 0

∫e^(2x) (√((1 −e^(2x) )))dx

$$\int{e}^{\mathrm{2x}} \sqrt{\left(\mathrm{1}\:−{e}^{\mathrm{2}{x}} \right)}{dx} \\ $$

Question Number 162054 Answers: 1 Comments: 0

Question Number 162073 Answers: 3 Comments: 0

prove that Ω =∫_(−∞) ^( +∞) (( cos (x))/((2+ 2x +x^( 2) )^( 2) )) dx = (π/e) cos(1)

$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\Omega\:=\int_{−\infty} ^{\:+\infty} \frac{\:{cos}\:\left({x}\right)}{\left(\mathrm{2}+\:\mathrm{2}{x}\:+{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{{e}}\:{cos}\left(\mathrm{1}\right) \\ $$

Question Number 162016 Answers: 2 Comments: 2

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

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\mathrm{3x}\right)}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 162015 Answers: 2 Comments: 0

find ∫_(−∞) ^(+∞) (dx/((x^2 −x+1)^4 ))

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

Question Number 162002 Answers: 1 Comments: 0

calculate Ω = ∫_0 ^( 1) Li_( 2) (1 − x^( 4) )dx = ? −−−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:{calculate} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{Li}_{\:\mathrm{2}} \:\left(\mathrm{1}\:−\:{x}^{\:\mathrm{4}} \right){dx}\:=\:? \\ $$$$\:\:\:\:−−−−− \\ $$

Question Number 161994 Answers: 0 Comments: 0

∫_0 ^1 ((ln∣x∣ln∣((1+x)/(1−x))∣)/(1−x^2 ))dx=???

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}\mid\boldsymbol{{x}}\mid\boldsymbol{\mathrm{ln}}\mid\frac{\mathrm{1}+\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}}\mid}{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=??? \\ $$

Question Number 161967 Answers: 3 Comments: 0

∫_0 ^1 ((ln^2 (x))/((1−x^2 )))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{x}}\right)}{\left(\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 161966 Answers: 1 Comments: 0

∫x^2 7^x^2 dx=?

$$\int\boldsymbol{\mathrm{x}}^{\mathrm{2}} \mathrm{7}^{\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 161919 Answers: 1 Comments: 0

∫_(−∞) ^( ∞) sin(x^2 +x+1)dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{−\infty} ^{\:\infty} \:\mathrm{sin}\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right){dx}\: \\ $$$$\: \\ $$

Question Number 161917 Answers: 0 Comments: 0

∫_(−∞) ^( ∞) (1/( (√(x^4 +x+1)) )) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{−\infty} ^{\:\infty} \:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{4}} +{x}+\mathrm{1}}\:}\:{dx} \\ $$$$\: \\ $$

Question Number 161839 Answers: 3 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +2x+2)^2 ))

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

Question Number 161830 Answers: 1 Comments: 0

∫(dx/( (√(1−x^(14) ))))

$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{14}} }} \\ $$

Question Number 161773 Answers: 3 Comments: 1

Question Number 161706 Answers: 1 Comments: 0

J =∫_0 ^( 1) (( 1−x)/(( 1+x +x^( 2) + x^( 3) )ln(x))) dx=?

$$\: \\ $$$$\mathrm{J}\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{1}−{x}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} +\:{x}^{\:\mathrm{3}} \:\right){ln}\left({x}\right)}\:{dx}=? \\ $$$$ \\ $$

Question Number 161660 Answers: 3 Comments: 0

∫_0 ^(π/4) ln(1+(√2)cos(x))dx=???

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \boldsymbol{{ln}}\left(\mathrm{1}+\sqrt{\mathrm{2}}\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)\boldsymbol{{dx}}=??? \\ $$

Question Number 161656 Answers: 1 Comments: 3

∫_0 ^1 ((xln(1+x))/(1+x^2 ))dx

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

Question Number 161652 Answers: 0 Comments: 0

∫_0 ^( 1) ((xlog(a+x))/(1+x^2 ))dx ∀ ∣a∣ ∈ N

$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{xlog}\left(\mathrm{a}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:\forall\:\mid\mathrm{a}\mid\:\in\:\mathbb{N} \\ $$

Question Number 161703 Answers: 2 Comments: 1

(1)∫ ((sin x−cos x)/( (√(sin 2x)))) dx (2) ∫_0 ^( π/2) cos 7x cos 17x cos 37x dx

$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{sin}\:\mathrm{2}{x}}}\:{dx} \\ $$$$\left(\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{cos}\:\mathrm{7}{x}\:\mathrm{cos}\:\mathrm{17}{x}\:\mathrm{cos}\:\mathrm{37}{x}\:{dx} \\ $$

Question Number 161646 Answers: 0 Comments: 0

Question Number 161609 Answers: 0 Comments: 0

∫_0 ^1 ((xln(1+x^4 ))/(1+x^2 ))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{{xln}}\left(\mathrm{1}+\boldsymbol{{x}}^{\mathrm{4}} \right)}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$$$ \\ $$

Question Number 161537 Answers: 2 Comments: 0

∫_0 ^( (π/4)) ((1+tan^4 (x))/(cot^2 (x))) dx =?

$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{1}+\mathrm{tan}\:^{\mathrm{4}} \left({x}\right)}{\mathrm{cot}\:^{\mathrm{2}} \left({x}\right)}\:{dx}\:=? \\ $$

Pg 54 Pg 55 Pg 56 Pg 57 Pg 58 Pg 59 Pg 60 Pg 61 Pg 62 Pg 63

Terms of Service

Contact: info@tinkutara.com