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IntegrationQuestion and Answers: Page 118

Question Number 127341 Answers: 0 Comments: 0

Question Number 127464 Answers: 1 Comments: 0

Question Number 127237 Answers: 5 Comments: 1

Nice...∫ ((√(1−ln^2 (x)))/(x ln (x))) dx ∫ (√(x/(1−x^3 ))) dx ∫ (√((4−x)/x)) dx

$$\:{Nice}...\int\:\frac{\sqrt{\mathrm{1}−\mathrm{ln}\:^{\mathrm{2}} \left({x}\right)}}{{x}\:\mathrm{ln}\:\left({x}\right)}\:{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\sqrt{\frac{{x}}{\mathrm{1}−{x}^{\mathrm{3}} }}\:{dx}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\sqrt{\frac{\mathrm{4}−{x}}{{x}}}\:{dx}\: \\ $$

Question Number 127236 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((lnx)/((x^2 −x+1)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{lnx}}{\left({x}^{\mathrm{2}} \:−{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 127224 Answers: 2 Comments: 0

... calculus (I) −complex analysis... calculate :: Φ = ∫_0 ^( ∞) ((ln(x))/(x^2 +3x+2)) dx=((ln^2 (2))/2)

$$\:\:...\:{calculus}\:\:\left({I}\right)\:−{complex}\:{analysis}... \\ $$$$\:\:\:\:{calculate}\:::\: \\ $$$$\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}}\:{dx}=\frac{{ln}^{\mathrm{2}} \left(\mathrm{2}\right)}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 127190 Answers: 2 Comments: 0

∫ (((√a)−(√x))/(1−(√(ax)))) dx =? ; a>0

$$\:\int\:\frac{\sqrt{{a}}−\sqrt{{x}}}{\mathrm{1}−\sqrt{{ax}}}\:{dx}\:=?\:;\:{a}>\mathrm{0} \\ $$

Question Number 171744 Answers: 1 Comments: 0

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

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

Question Number 127157 Answers: 2 Comments: 0

D={(x,y):∣x∣+∣y∣≤2} ∫∫_D e^(x+y) dydx=?

$${D}=\left\{\left({x},{y}\right):\mid{x}\mid+\mid{y}\mid\leqslant\mathrm{2}\right\} \\ $$$$\int\underset{{D}} {\int}{e}^{{x}+{y}} {dydx}=? \\ $$

Question Number 127161 Answers: 1 Comments: 0

R=(x,y):y≥0 , x^2 +y^2 ≤9} ∫∫_R cos(x^2 +y^2 )dydx=?

$$\left.{R}=\left({x},{y}\right):{y}\geqslant\mathrm{0}\:,\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \leqslant\mathrm{9}\right\} \\ $$$$\int\underset{{R}} {\int}{cos}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right){dydx}=? \\ $$

Question Number 127160 Answers: 1 Comments: 0

R ={(x,y): (x−2)^2 +y^2 ≤4} ∫∫_R (x^2 +y^2 )^2 dydx=?

$${R}\:=\left\{\left({x},{y}\right):\:\left({x}−\mathrm{2}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} \leqslant\mathrm{4}\right\} \\ $$$$\int\underset{{R}} {\int}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\mathrm{2}} {dydx}=? \\ $$

Question Number 127110 Answers: 1 Comments: 0

∫ (arcsin x)^2 dx =?

$$\:\:\int\:\left(\mathrm{arcsin}\:{x}\right)^{\mathrm{2}} \:{dx}\:=? \\ $$

Question Number 127042 Answers: 2 Comments: 1

∫_(1/(√2)) ^( 1) ((arcsin x)/x^3 ) dx ? ′ not nice integral ′

$$\:\int_{\mathrm{1}/\sqrt{\mathrm{2}}} ^{\:\mathrm{1}} \frac{\mathrm{arcsin}\:{x}}{{x}^{\mathrm{3}} }\:{dx}\:? \\ $$$$\:'\:{not}\:{nice}\:{integral}\:'\: \\ $$

Question Number 127032 Answers: 2 Comments: 0

Question Number 127020 Answers: 3 Comments: 1

super nice ! show that ζ(6) = (π^6 /(945))

$$\:\:{super}\:{nice}\:! \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{show}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\zeta\left(\mathrm{6}\right)\:=\:\frac{\pi^{\mathrm{6}} }{\mathrm{945}} \\ $$

Question Number 127017 Answers: 2 Comments: 0

...NICE CALCULUS... prove that :: ∫_0 ^( ∞) (((x^2 ln(πx))/π^(πx) ))dx =(1/((πln(π))^3 ))[(3−2(γ+ln(ln(π)))]

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{NICE}\:\:\:\:\:{CALCULUS}... \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\left(\frac{{x}^{\mathrm{2}} {ln}\left(\pi{x}\right)}{\pi^{\pi{x}} }\right){dx} \\ $$$$\:\:=\frac{\mathrm{1}}{\left(\pi{ln}\left(\pi\right)\right)^{\mathrm{3}} }\left[\left(\mathrm{3}−\mathrm{2}\left(\gamma+{ln}\left({ln}\left(\pi\right)\right)\right)\right]\right. \\ $$

Question Number 126997 Answers: 1 Comments: 0

∫_0 ^1 arcsin (((sin x)/( (√2)))) dx =?

$$\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{arcsin}\:\left(\frac{\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{2}}}\right)\:{dx}\:=? \\ $$

Question Number 126986 Answers: 1 Comments: 0

... nice calculus... prove that :: I := ∫_0 ^( (π/2)) (({cot(x)})/(cot(x)))dx=(1/2)(π−ln(((sinh(π))/π))) {x} is fractional part of x ..

$$\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\left\{{cot}\left({x}\right)\right\}}{{cot}\left({x}\right)}{dx}=\frac{\mathrm{1}}{\mathrm{2}}\left(\pi−{ln}\left(\frac{{sinh}\left(\pi\right)}{\pi}\right)\right) \\ $$$$\left\{{x}\right\}\:{is}\:{fractional}\:{part}\:{of}\:\:{x}\:.. \\ $$

Question Number 126879 Answers: 2 Comments: 0

∫ (x^4 /(1+x^8 )) dx ?

$$\:\int\:\frac{{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{8}} }\:{dx}\:? \\ $$

Question Number 126873 Answers: 3 Comments: 0

calculate ∫_(2019) ^(2021) (dx/((x−1)^(2019) (x+1)^(2021) ))

$$\mathrm{calculate}\:\int_{\mathrm{2019}} ^{\mathrm{2021}} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{2019}} \left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2021}} } \\ $$

Question Number 126865 Answers: 2 Comments: 4

∫_0 ^( π) (x/(2+cos(2x)))dx = 0 Prove or Disprove

$$\int_{\mathrm{0}} ^{\:\pi} \frac{\mathrm{x}}{\mathrm{2}+\mathrm{cos}\left(\mathrm{2x}\right)}\mathrm{dx}\:=\:\mathrm{0} \\ $$$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{Disprove} \\ $$

Question Number 127725 Answers: 1 Comments: 0

... advanced mathematics... prove that :: Σ_(n=0) ^∞ (1/(2^n (((3n)),(( n)) ))) =^(???) (3/(125))(((11π)/6)−2log(2)+45)

$$\:\:\:\:\:\:\:...\:{advanced}\:\:{mathematics}... \\ $$$$\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{{n}} \begin{pmatrix}{\mathrm{3}{n}}\\{\:\:{n}}\end{pmatrix}}\:\overset{???} {=}\frac{\mathrm{3}}{\mathrm{125}}\left(\frac{\mathrm{11}\pi}{\mathrm{6}}−\mathrm{2}{log}\left(\mathrm{2}\right)+\mathrm{45}\right) \\ $$$$ \\ $$

Question Number 127726 Answers: 0 Comments: 0

...nice calculus... evaluate:: Ω=∫_0 ^( (π/2)) ∫_0 ^( (π/2)) (((ln(cos((x/2)))−ln(cos((y/2))))/(cos(x)−cos(y))))dxdy

$$\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:{evaluate}:: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\left(\frac{{ln}\left({cos}\left(\frac{{x}}{\mathrm{2}}\right)\right)−{ln}\left({cos}\left(\frac{{y}}{\mathrm{2}}\right)\right)}{{cos}\left({x}\right)−{cos}\left({y}\right)}\right){dxdy} \\ $$$$ \\ $$

Question Number 126803 Answers: 2 Comments: 0

B((7/3),(2/3)) =? B = betha function

$$\:\:{B}\left(\frac{\mathrm{7}}{\mathrm{3}},\frac{\mathrm{2}}{\mathrm{3}}\right)\:=? \\ $$$${B}\:=\:{betha}\:{function}\: \\ $$

Question Number 126788 Answers: 4 Comments: 0

σ = ∫_0 ^( ∞) (√x) e^(−x/4) dx = ?

$$\:\sigma\:=\:\underset{\mathrm{0}} {\overset{\:\:\:\:\:\infty} {\int}}\sqrt{{x}}\:{e}^{−{x}/\mathrm{4}} \:{dx}\:=\:?\: \\ $$

Question Number 126766 Answers: 4 Comments: 0

...calculus (I)... please evaluate :: Ψ=∫_0 ^( 1) ( (1/(1+x^6 )) )dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{calculus}\:\:\left({I}\right)... \\ $$$$\:\:\:\:{please}\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Psi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{6}} }\:\right){dx}=? \\ $$$$ \\ $$

Question Number 126753 Answers: 1 Comments: 0

∫ ((sin x+cos x)/(3sin x+4cos x+1)) dx

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

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