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IntegrationQuestion and Answers: Page 118

Question Number 128826 Answers: 1 Comments: 0

∫_(−1) ^( 5) (√((2x^2 −8)/x)) dx =?

$$\int_{−\mathrm{1}} ^{\:\mathrm{5}} \sqrt{\frac{\mathrm{2x}^{\mathrm{2}} −\mathrm{8}}{\mathrm{x}}}\:\mathrm{dx}\:=? \\ $$

Question Number 128797 Answers: 2 Comments: 0

...nice calculus... φ =^(???) ∫_0 ^( ∞) (((tanh(x))/e^x )) dx

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\phi\:\overset{???} {=}\int_{\mathrm{0}} ^{\:\infty} \left(\frac{{tanh}\left({x}\right)}{{e}^{{x}} }\right)\:{dx} \\ $$$$ \\ $$

Question Number 128775 Answers: 3 Comments: 0

∫ (dx/((1−x)^2 (√(1−x^2 )))) ?

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

Question Number 128750 Answers: 1 Comments: 1

Given a function f satisfy f(−x)=3f(x). If ∫_(−1) ^( 2) f(x) dx = 2 then ∫_(−2) ^( 1) f(x)dx=?

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{function}\:\mathrm{f}\:\mathrm{satisfy}\:\mathrm{f}\left(−\mathrm{x}\right)=\mathrm{3f}\left(\mathrm{x}\right). \\ $$$$\mathrm{If}\:\int_{−\mathrm{1}} ^{\:\mathrm{2}} \mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{2}\:\mathrm{then}\:\int_{−\mathrm{2}} ^{\:\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 128736 Answers: 1 Comments: 1

... nice calculus... evluate :: φ = ∫_0 ^( ∞) e^(−x^2 ) cos(x)dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:{evluate}\::: \\ $$$$\:\:\:\:\:\phi\:=\:\int_{\mathrm{0}} ^{\:\infty} {e}^{−{x}^{\mathrm{2}} } {cos}\left({x}\right){dx}=? \\ $$$$ \\ $$

Question Number 128721 Answers: 0 Comments: 1

∫_0 ^1 ((ln x)/(x(x^2 +1))) dx

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

Question Number 128710 Answers: 0 Comments: 0

∫e^x (((1+sinx+cosx)/(cos^2 x))) dx

$$\int{e}^{{x}} \left(\frac{\mathrm{1}+{sinx}+{cosx}}{{cos}^{\mathrm{2}} {x}}\right)\:{dx} \\ $$

Question Number 128707 Answers: 1 Comments: 0

...nice calculus... prove that:: ∫_0 ^( ∞) ((ln(1+ϕ^2 x^2 ))/(1+π^2 x^2 )) dx=ln(((π+ϕ)/π)) ϕ::= golen ratio...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\mathrm{nice}\:\:\mathrm{calculus}... \\ $$$${prove}\:\:{that}::\: \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left(\mathrm{1}+\varphi^{\mathrm{2}} {x}^{\mathrm{2}} \right)}{\mathrm{1}+\pi^{\mathrm{2}} {x}^{\mathrm{2}} }\:{dx}={ln}\left(\frac{\pi+\varphi}{\pi}\right) \\ $$$$\varphi::=\:\:{golen}\:{ratio}... \\ $$$$ \\ $$

Question Number 128702 Answers: 1 Comments: 0

If ((sin^4 x)/2) + ((cos^4 x)/3) = (1/5) then ((sin^8 x)/8) + ((cos^8 x)/(27)) = ?

$$\:\mathrm{If}\:\frac{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}}{\mathrm{2}}\:+\:\frac{\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}{\mathrm{3}}\:=\:\frac{\mathrm{1}}{\mathrm{5}}\:\mathrm{then}\: \\ $$$$\:\frac{\mathrm{sin}\:^{\mathrm{8}} \mathrm{x}}{\mathrm{8}}\:+\:\frac{\mathrm{cos}\:^{\mathrm{8}} \mathrm{x}}{\mathrm{27}}\:=\:? \\ $$

Question Number 128680 Answers: 1 Comments: 0

...nice calculus... Σ_(n=0) ^∞ (1/((3n+1)ϕ^(3n+1) )) =? ϕ :: golden ratio...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{3}{n}+\mathrm{1}\right)\varphi^{\mathrm{3}{n}+\mathrm{1}} }\:=? \\ $$$$\varphi\:::\:\:{golden}\:\:{ratio}... \\ $$$$ \\ $$

Question Number 128664 Answers: 2 Comments: 0

∫_0 ^( π/2) (1−sin x+sin^2 x−sin^3 x+sin^4 x−sin^5 x+...) dx =?

$$\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \left(\mathrm{1}−\mathrm{sin}\:\mathrm{x}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}−\mathrm{sin}\:^{\mathrm{5}} \mathrm{x}+...\right)\:\mathrm{dx}\:=? \\ $$

Question Number 128634 Answers: 1 Comments: 0

θ = ∫ (1+4x^4 )e^x^4 dx

$$\theta\:=\:\int\:\left(\mathrm{1}+\mathrm{4x}^{\mathrm{4}} \right)\mathrm{e}^{\mathrm{x}^{\mathrm{4}} } \:\mathrm{dx}\: \\ $$

Question Number 128633 Answers: 2 Comments: 0

Ω = ∫_0 ^( (1/3)) x^(2n) ln(1−x)dx

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

Question Number 128620 Answers: 1 Comments: 0

Question Number 128610 Answers: 1 Comments: 1

∫_(−π/4) ^( π/4) ((sec x)/(e^x +1)) dx

$$\int_{−\pi/\mathrm{4}} ^{\:\pi/\mathrm{4}} \frac{\mathrm{sec}\:\mathrm{x}}{\mathrm{e}^{\mathrm{x}} +\mathrm{1}}\:\mathrm{dx}\: \\ $$

Question Number 128608 Answers: 1 Comments: 0

∫ x^2 .tan^(−1) ((x/2))dx=?

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

Question Number 128602 Answers: 1 Comments: 0

∫_(−1) ^1 ∫_0 ^(1−x) (√((x^(2/3) y−x^(5/3) y−x^(2/3) y^2 )/y^2 ))dydx

$$\int_{−\mathrm{1}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}−{x}} \sqrt{\frac{{x}^{\frac{\mathrm{2}}{\mathrm{3}}} {y}−{x}^{\frac{\mathrm{5}}{\mathrm{3}}} {y}−{x}^{\frac{\mathrm{2}}{\mathrm{3}}} {y}^{\mathrm{2}} }{{y}^{\mathrm{2}} }}{dydx} \\ $$$$ \\ $$

Question Number 128575 Answers: 3 Comments: 3

∫(√(x^2 +4x+13))dx=??

$$\int\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{13}}{dx}=?? \\ $$

Question Number 128570 Answers: 0 Comments: 0

... mathematical analysis... if ′′ f ′′ is Reimann integrable function on [a , b ] , then prove:: lim_(t→∞ ) {∫_a ^( b) f(x)cos(tx)dx }=0 ..Reimann−Lebesgue theorem...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{mathematical}\:\:{analysis}... \\ $$$$\:\:{if}\:''\:\:{f}\:\:\:''\:\:{is}\:\mathscr{R}{eimann}\:{integrable} \\ $$$$\:\:\:{function}\:\:{on}\:\left[{a}\:,\:{b}\:\right]\:,\:{then}\:{prove}:: \\ $$$$\:\:\:\:\: \\ $$$$\:\:{lim}_{{t}\rightarrow\infty\:} \left\{\int_{{a}} ^{\:{b}} {f}\left({x}\right){cos}\left({tx}\right){dx}\:\right\}=\mathrm{0} \\ $$$$\:\:..\mathscr{R}{eimann}−\mathscr{L}{ebesgue}\:\:{theorem}... \\ $$$$ \\ $$

Question Number 128542 Answers: 1 Comments: 0

∫ (((x^4 −x)^(1/4) )/x^5 ) dx =?

$$\:\int\:\frac{\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}\right)^{\mathrm{1}/\mathrm{4}} }{\mathrm{x}^{\mathrm{5}} }\:\mathrm{dx}\:=? \\ $$

Question Number 128540 Answers: 1 Comments: 1

If f(x)=lim_(x→∞) ((x^n −x^(−n) )/(x^n +x^(−n) )) ,x>1 then ∫ ((xf(x) ln (x+(√(1+x^2 )) ))/( (√(1+x^2 )))) dx =?

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{n}} −\mathrm{x}^{−\mathrm{n}} }{\mathrm{x}^{\mathrm{n}} +\mathrm{x}^{−\mathrm{n}} }\:,\mathrm{x}>\mathrm{1} \\ $$$$\mathrm{then}\:\int\:\frac{\mathrm{xf}\left(\mathrm{x}\right)\:\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\right)}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=? \\ $$

Question Number 128529 Answers: 1 Comments: 2

Question Number 128511 Answers: 1 Comments: 0

H=∫ ((2017x^(2016) +2018x^(2017) )/(1+x^(4034) +2x^(4035) +x^(4036) )) dx

$$\:\mathcal{H}=\int\:\frac{\mathrm{2017x}^{\mathrm{2016}} +\mathrm{2018x}^{\mathrm{2017}} }{\mathrm{1}+\mathrm{x}^{\mathrm{4034}} +\mathrm{2x}^{\mathrm{4035}} +\mathrm{x}^{\mathrm{4036}} }\:\mathrm{dx}\: \\ $$

Question Number 128499 Answers: 1 Comments: 0

find u_n =∫_1 ^∞ (([ne^(−x) ])/n^3 )dx

$$\mathrm{find}\:\:\mathrm{u}_{\mathrm{n}} =\int_{\mathrm{1}} ^{\infty} \:\:\frac{\left[\mathrm{ne}^{−\mathrm{x}} \right]}{\mathrm{n}^{\mathrm{3}} }\mathrm{dx} \\ $$

Question Number 128495 Answers: 0 Comments: 0

find ∫_0 ^∞ ((sinx)/([x]))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{sinx}}{\left[\mathrm{x}\right]}\mathrm{dx} \\ $$

Question Number 128471 Answers: 2 Comments: 0

... calculus ... Φ=^? ∫_0 ^( 1) (ln(x))^2 ln((√(−ln(x))) dx

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}\:... \\ $$$$\:\:\:\:\:\:\:\Phi\overset{?} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({ln}\left({x}\right)\right)^{\mathrm{2}} {ln}\left(\sqrt{−{ln}\left({x}\right)}\:{dx}\right. \\ $$

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