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IntegrationQuestion and Answers: Page 120

Question Number 127432 Answers: 1 Comments: 0

...CALCULUS... ∅=∫_(−∞) ^( +∞) cos(((πx^2 )/2))dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\mathrm{CALCULUS}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\varnothing=\int_{−\infty} ^{\:+\infty} {cos}\left(\frac{\pi{x}^{\mathrm{2}} }{\mathrm{2}}\right){dx}=? \\ $$$$ \\ $$

Question Number 127420 Answers: 1 Comments: 0

∫_0 ^( ∞) ((x−1)/( (√(2^x −1)) ln (2^x −1))) dx ?

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

Question Number 127405 Answers: 0 Comments: 0

μ = ∫ (x^7 /(x^(10) +169)) dx

$$\:\mu\:=\:\int\:\frac{{x}^{\mathrm{7}} }{{x}^{\mathrm{10}} +\mathrm{169}}\:{dx}\: \\ $$

Question Number 127377 Answers: 0 Comments: 2

Question Number 127370 Answers: 1 Comments: 1

... advanced calculus .. prove:: ((1023)/(134))∫_0 ^( ∞) ((x^(2/5) +x^((−2)/5) )/((1+x^2 )(1+1024x^2 )))dx=(π/ϕ) ϕ: golden ratio...

$$\:\:\:\:\:\:\:\:\:...\:\:{advanced}\:\:{calculus}\:\:.. \\ $$$$\:\:{prove}:: \\ $$$$\:\:\:\frac{\mathrm{1023}}{\mathrm{134}}\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{\frac{\mathrm{2}}{\mathrm{5}}} +{x}^{\frac{−\mathrm{2}}{\mathrm{5}}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{1024}{x}^{\mathrm{2}} \right)}{dx}=\frac{\pi}{\varphi} \\ $$$$\:\:\:\varphi:\:{golden}\:\:{ratio}... \\ $$$$ \\ $$

Question Number 127368 Answers: 1 Comments: 0

∫ ((√(tan x))/( (√(tan^2 x − 1)))) dx

$$\int\:\frac{\sqrt{\mathrm{tan}\:\mathrm{x}}}{\:\sqrt{\mathrm{tan}^{\mathrm{2}} \mathrm{x}\:\:−\:\:\mathrm{1}}}\:\:\mathrm{dx} \\ $$

Question Number 127355 Answers: 1 Comments: 0

∫ (dx/((1+x^2 )(√(1−x^2 )))) =?

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

Question Number 127350 Answers: 1 Comments: 0

∫ (x^2 /((x^2 −1)^(5/2) )) dx ?

$$\:\int\:\frac{{x}^{\mathrm{2}} }{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{5}}{\mathrm{2}}} }\:{dx}\:? \\ $$

Question Number 127349 Answers: 1 Comments: 1

∫ (((1−s^2 )^(5/2) )/s^8 ) ds =?

$$\:\int\:\frac{\left(\mathrm{1}−{s}^{\mathrm{2}} \right)^{\mathrm{5}/\mathrm{2}} }{{s}^{\mathrm{8}} }\:{ds}\:=? \\ $$

Question Number 127435 Answers: 0 Comments: 0

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 =?

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

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