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IntegrationQuestion and Answers: Page 59

Question Number 160597 Answers: 1 Comments: 0

∫_0 ^( (π/2)) (dx/(2−cos x)) =?

$$\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{dx}}{\mathrm{2}−\mathrm{cos}\:\mathrm{x}}\:=? \\ $$

Question Number 160594 Answers: 1 Comments: 0

∫ (dx/(1−tan^2 (x))) =?

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

Question Number 160590 Answers: 0 Comments: 0

Ω:=∫_0 ^( (1/2)) (( arcsinh(x))/x) dx =^? (π^2 /(20))

$$ \\ $$$$\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{\:{arcsinh}\left({x}\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\pi^{\mathrm{2}} }{\mathrm{20}} \\ $$

Question Number 160563 Answers: 1 Comments: 0

∫x{x}[x]dx=?

$$\int\mathrm{x}\left\{\mathrm{x}\right\}\left[\mathrm{x}\right]\mathrm{dx}=? \\ $$

Question Number 160556 Answers: 0 Comments: 0

∫_0 ^∞ ((arctg(x))/(1+x))∙(dx/( (x)^(1/4) ))=?

$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{arctg}}\left(\boldsymbol{\mathrm{x}}\right)}{\mathrm{1}+\boldsymbol{\mathrm{x}}}\centerdot\frac{\boldsymbol{\mathrm{dx}}}{\:\sqrt[{\mathrm{4}}]{\boldsymbol{\mathrm{x}}}}=? \\ $$

Question Number 160551 Answers: 1 Comments: 2

∫_0 ^1 arctan x∙ln(1+x)dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{arctan}\:\mathrm{x}\centerdot\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 160547 Answers: 1 Comments: 0

solve Ω=∫_0 ^( ∞) (( tan^( −1) ( x ))/((1+ x^( 2) )(√( x)))) dx= ? −−−−−−−−

$$\:\:{solve} \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \left(\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \:\right)\sqrt{\:{x}}}\:{dx}=\:? \\ $$$$−−−−−−−− \\ $$

Question Number 160543 Answers: 0 Comments: 0

lim_( x→ 6) (( Γ ( sin( (π/x)))−Γ ((3/x) ))/(sin( πx )))= ?

$$ \\ $$$$\mathrm{lim}_{\:{x}\rightarrow\:\mathrm{6}} \frac{\:\Gamma\:\left(\:{sin}\left(\:\frac{\pi}{{x}}\right)\right)−\Gamma\:\left(\frac{\mathrm{3}}{{x}}\:\right)}{{sin}\left(\:\pi{x}\:\right)}=\:? \\ $$$$ \\ $$

Question Number 160415 Answers: 1 Comments: 0

∫_0 ^∞ (t^n /(1+t+t^2 ))dt=?

$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{\mathrm{t}}^{\boldsymbol{\mathrm{n}}} }{\mathrm{1}+\boldsymbol{\mathrm{t}}+\boldsymbol{\mathrm{t}}^{\mathrm{2}} }\boldsymbol{\mathrm{dt}}=? \\ $$

Question Number 160362 Answers: 0 Comments: 0

∫(x^n /( (√(x−x^2 ))))dx=?

$$\int\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{n}}} }{\:\sqrt{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 160358 Answers: 1 Comments: 0

∫_0 ^(π/2) ln(sinx)ln(cosx)dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{sin}{x}\right)\mathrm{ln}\left(\mathrm{cos}{x}\right){dx} \\ $$

Question Number 160349 Answers: 1 Comments: 0

find ∫(dx/(x+e^x ))=?

$${find}\:\int\frac{{dx}}{{x}+{e}^{{x}} }=? \\ $$

Question Number 160328 Answers: 0 Comments: 0

∫_1 ^( 2e) (√(((x^2 +1)/(ln x)) (√(ln x^2 )))) dx =?

$$\:\int_{\mathrm{1}} ^{\:\mathrm{2e}} \:\sqrt{\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{ln}\:\mathrm{x}}\:\sqrt{\mathrm{ln}\:\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=? \\ $$

Question Number 160291 Answers: 3 Comments: 0

Ω = ∫ ((x^2 −3x+7)/((x^2 −4x+6)^2 )) dx =?

$$\:\:\:\Omega\:=\:\int\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{7}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4x}+\mathrm{6}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 160281 Answers: 1 Comments: 0

∫_0 ^1 ((ln^2 (1−x)lnx)/x)dx=?

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

Question Number 160276 Answers: 1 Comments: 0

∫_0 ^1 ((ln(1−x)ln(x))/(1−x))dx=?

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

Question Number 160204 Answers: 1 Comments: 1

∫_0 ^1 (x/(1−x^3 ))=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{3}} }=? \\ $$

Question Number 160194 Answers: 1 Comments: 1

∫_0 ^1 (x/(1−x^6 ))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{x}}}{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{6}} }\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 160113 Answers: 1 Comments: 0

prove Φ:= ∫_0 ^( ∞) (( sinh(x))/(cosh^2 (x))).(1/x) dx =^(???) ((4G)/π)

$$ \\ $$$$\:\:{prove}\:\: \\ $$$$\:\:\:\:\:\:\Phi:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sinh}\left({x}\right)}{{cosh}^{\mathrm{2}} \left({x}\right)}.\frac{\mathrm{1}}{{x}}\:{dx}\:\overset{???} {=}\:\frac{\mathrm{4G}}{\pi} \\ $$

Question Number 160013 Answers: 0 Comments: 1

∫(1/(4sin x+3cos x))dx evaluate

$$\int\frac{\mathrm{1}}{\mathrm{4}{sin}\:{x}+\mathrm{3}{cos}\:{x}}{dx} \\ $$$${evaluate} \\ $$

Question Number 159960 Answers: 1 Comments: 0

∫_0 ^( 1) ((3x^3 −x^2 +2x−4)/( (√(x^2 −3x+2)))) dx =?

$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{4}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}}\:{dx}\:=?\: \\ $$

Question Number 159931 Answers: 0 Comments: 0

Prove :: ∫_0 ^∞ ((sin^n x)/x^m )dx=(1/(Γ(m)))∫_0 ^∞ ((D^(m−1) sin^n x)/x)dx n+m∈Odd Number.

$$\mathrm{Prove}\:::\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}^{\mathrm{n}} \mathrm{x}}{\mathrm{x}^{\mathrm{m}} }\mathrm{dx}=\frac{\mathrm{1}}{\Gamma\left(\mathrm{m}\right)}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{D}^{\mathrm{m}−\mathrm{1}} \mathrm{sin}^{\mathrm{n}} \mathrm{x}}{\mathrm{x}}\mathrm{dx} \\ $$$$\mathrm{n}+\mathrm{m}\in\mathrm{Odd}\:\mathrm{Number}. \\ $$

Question Number 159917 Answers: 0 Comments: 1

Question Number 159915 Answers: 0 Comments: 1

∫_1 ^(16) ((√x)/(1+(x^3 )^(1/4) )) dx =?

$$\:\:\int_{\mathrm{1}} ^{\mathrm{16}} \:\frac{\sqrt{{x}}}{\mathrm{1}+\sqrt[{\mathrm{4}}]{{x}^{\mathrm{3}} }}\:{dx}\:=? \\ $$

Question Number 159870 Answers: 1 Comments: 0

∫ ((1−cot^2 x)/(1+sin x)) dx =?

$$\:\:\int\:\frac{\mathrm{1}−\mathrm{cot}\:^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx}\:=? \\ $$

Question Number 159823 Answers: 2 Comments: 0

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