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IntegrationQuestion and Answers: Page 126

Question Number 123454 Answers: 1 Comments: 0

...nice calculus... calculate ::: Ω =^(???) ∫_0 ^( ∞) (√x) Π_(n=1) ^∞ (cos((x/2^n )))dx

$$\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{calculate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:\overset{???} {=}\int_{\mathrm{0}} ^{\:\infty} \sqrt{{x}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left({cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)\right){dx} \\ $$

Question Number 123526 Answers: 3 Comments: 2

∫_0 ^∞ ((x arctan x)/((1+x^2 )^2 )) dx ?

$$\:\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{x}\:\mathrm{arctan}\:{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:? \\ $$

Question Number 123387 Answers: 0 Comments: 2

... nice calculus... prove that :: Ω=∫_0 ^( 1) (((ln(x))^2 li_3 (x))/(1−x)) dx =^(???) ζ^2 (3)−ζ(6) ✓

$$\:\:\:\:\:...\:\:{nice}\:{calculus}... \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({ln}\left({x}\right)\right)^{\mathrm{2}} {li}_{\mathrm{3}} \left({x}\right)}{\mathrm{1}−{x}}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{???} {=}\zeta^{\mathrm{2}} \left(\mathrm{3}\right)−\zeta\left(\mathrm{6}\right)\:\checkmark \\ $$

Question Number 123386 Answers: 1 Comments: 0

Given f(x)=(∫_0 ^1 f(x)dx)x^2 +(∫_0 ^2 f(x)dx)x+(∫_0 ^3 f(x)dx)+1 then the value of f(4) = ...

$$\:{Given}\: \\ $$$${f}\left({x}\right)=\left(\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} +\left(\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}\right){x}+\left(\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}\right)+\mathrm{1} \\ $$$${then}\:{the}\:{value}\:{of}\:{f}\left(\mathrm{4}\right)\:=\:... \\ $$

Question Number 123361 Answers: 0 Comments: 1

please find the ∫(e^x /x)dx

$${please}\:{find}\:{the}\:\int\frac{{e}^{{x}} }{{x}}{dx} \\ $$

Question Number 123352 Answers: 0 Comments: 0

∫(e^x /x)dx

$$\int\frac{{e}^{{x}} }{{x}}{dx} \\ $$

Question Number 123331 Answers: 1 Comments: 0

∫ ((sinx)/x) dx

$$\int\:\frac{{sinx}}{{x}}\:{dx} \\ $$

Question Number 123261 Answers: 1 Comments: 0

... nice calculus... prove that:: Ω=∫_R e^(x−sinh^2 (x)) dx=(√π)

$$\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:{prove}\:\:{that}:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega=\int_{\mathbb{R}} {e}^{{x}−{sinh}^{\mathrm{2}} \left({x}\right)} {dx}=\sqrt{\pi} \\ $$

Question Number 123255 Answers: 1 Comments: 0

∗∗∗ nice calculus ∗∗∗ evaluate :: Φ=∫_0 ^(π/2) log^3 (tan(x))dx =?

$$\:\:\:\:\:\:\:\:\:\:\ast\ast\ast\:\:{nice}\:\:{calculus}\:\ast\ast\ast \\ $$$$\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({tan}\left({x}\right)\right){dx}\:=? \\ $$

Question Number 123253 Answers: 0 Comments: 0

Question Number 123234 Answers: 3 Comments: 1

∫ (√(x^2 −4x+5)) dx

$$\:\:\int\:\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{5}}\:{dx}\: \\ $$

Question Number 123177 Answers: 0 Comments: 0

lebesgue measure on [0 1] is finite ? true or false give reason

$${lebesgue}\:{measure}\:{on}\:\left[\mathrm{0}\:\mathrm{1}\right]\:{is}\:{finite}\:?\:{true}\:{or}\:{false}\:{give}\:{reason} \\ $$

Question Number 123159 Answers: 3 Comments: 0

∫_( 0) ^( (π/2)) log^2 (tan(x))dx

$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{log}^{\mathrm{2}} \left(\mathrm{tan}\left(\mathrm{x}\right)\right)\mathrm{dx} \\ $$

Question Number 123154 Answers: 3 Comments: 0

Question Number 123060 Answers: 3 Comments: 0

.... nice calculus .... evaluate ::: Ω=^(???) ∫_(−∞) ^( ∞) (x^2 /((1+e^x )(1+e^(−x) )))dx

$$\:\:\:\:\:\:\:\:\:\:....\:\:\:{nice}\:\:{calculus}\:.... \\ $$$$\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega\overset{???} {=}\int_{−\infty} ^{\:\infty} \frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{e}^{{x}} \right)\left(\mathrm{1}+{e}^{−{x}} \right)}{dx} \\ $$

Question Number 123037 Answers: 5 Comments: 0

∫ ((√(1−x))/(1−(√x))) dx

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

Question Number 123034 Answers: 1 Comments: 0

Question Number 123033 Answers: 2 Comments: 1

Evaluate the integral ∫_0 ^1 ((1−x^7 ))^(1/3) − ((1−x^3 ))^(1/7) dx .

$${Evaluate}\:{the}\:{integral}\: \\ $$$$\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt[{\mathrm{3}}]{\mathrm{1}−{x}^{\mathrm{7}} }\:−\:\sqrt[{\mathrm{7}}]{\mathrm{1}−{x}^{\mathrm{3}} }\:{dx}\:. \\ $$

Question Number 122980 Answers: 2 Comments: 0

∫_0 ^(ℓn 10) ((e^x (√(e^x −1)))/(e^x +8)) dx ?

$$\:\int_{\mathrm{0}} ^{\ell{n}\:\mathrm{10}} \:\frac{{e}^{{x}} \:\sqrt{{e}^{{x}} −\mathrm{1}}}{{e}^{{x}} +\mathrm{8}}\:{dx}\:? \\ $$

Question Number 122979 Answers: 1 Comments: 0

∫_0 ^π ((x sin x)/( (√(3+sin^2 x)))) dx ?

$$\:\:\int_{\mathrm{0}} ^{\pi} \frac{{x}\:\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{3}+\mathrm{sin}\:^{\mathrm{2}} {x}}}\:{dx}\:? \\ $$

Question Number 122976 Answers: 4 Comments: 0

∫_1 ^( ∞) ((tan^(−1) (x))/x^2 ) dx ?

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

Question Number 122967 Answers: 2 Comments: 0

∫ (x/( (√(1−x^4 )))) dx

$$\:\:\int\:\frac{{x}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{4}} }}\:{dx}\: \\ $$

Question Number 122963 Answers: 2 Comments: 0

∫_( 0) ^( 1) ((ln(1+x))/(1+x^2 ))dx

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

Question Number 123023 Answers: 1 Comments: 0

... advanced calculus... calculate::: I:=^(???) ∫_0 ^( π) (x/(1−sin(x)cos(x)))dx ................................

$$\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:\:{calculus}... \\ $$$${calculate}::: \\ $$$$\:\:\:\:\:\mathrm{I}:\overset{???} {=}\:\int_{\mathrm{0}} ^{\:\pi} \frac{{x}}{\mathrm{1}−{sin}\left({x}\right){cos}\left({x}\right)}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:................................ \\ $$

Question Number 123020 Answers: 1 Comments: 0

... advanced calculus.. calculate :: ∅=∫_0 ^( π) (π/(1−sin(x)cos(x)))dx=??? ....................

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:\:{calculus}.. \\ $$$$ \\ $$$$\:\:\:{calculate}\:::\:\:\:\emptyset=\int_{\mathrm{0}} ^{\:\pi} \frac{\pi}{\mathrm{1}−{sin}\left({x}\right){cos}\left({x}\right)}{dx}=??? \\ $$$$\:\:\:\:\:\:\:\:.................... \\ $$

Question Number 122942 Answers: 2 Comments: 1

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