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IntegrationQuestion and Answers: Page 126

Question Number 124587 Answers: 1 Comments: 0

...nice ◂::::▶ calculus simple question:: prove that :: ∫_0 ^( ∞) (4/( (√(4+x^4 )))) dx=^(???) ∫_0 ^( (π/2)) (dx/( (√(sin(x))))) +∫_0 ^( (π/2)) (dx/( (√(cos(x)))))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\blacktriangleleft::::\blacktriangleright\:{calculus} \\ $$$$\:\:\:\:\:{simple}\:\:{question}:: \\ $$$$\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{4}}{\:\sqrt{\mathrm{4}+{x}^{\mathrm{4}} }}\:{dx}\overset{???} {=}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\:\sqrt{{sin}\left({x}\right)}}\:+\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{dx}}{\:\sqrt{{cos}\left({x}\right)}} \\ $$

Question Number 124579 Answers: 5 Comments: 0

I=∫_0 ^( ∞) (dx/((x+(√(x^2 +1)))^2 )) ?

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

Question Number 124566 Answers: 4 Comments: 0

∫ (dx/((x+(√(x^2 +1)))^2 ))

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

Question Number 124542 Answers: 1 Comments: 1

help ∫3xdx

$${help}\:\:\int\mathrm{3}{xdx} \\ $$

Question Number 124540 Answers: 2 Comments: 0

∫((4x+9)/(x^2 +6x+10))dx

$$\int\frac{\mathrm{4}{x}+\mathrm{9}}{{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{10}}{dx} \\ $$

Question Number 124532 Answers: 5 Comments: 0

∫ (dx/(x^2 (√(x^2 −1)))) ?

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

Question Number 124531 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/((2x+1)^4 (x+3)^5 ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{4}} \left({x}+\mathrm{3}\right)^{\mathrm{5}} } \\ $$

Question Number 124530 Answers: 1 Comments: 1

find ∫_0 ^∞ cos(x^n )dx snd ∫_0 ^∞ sin(x^n )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{{n}} \right){dx}\:{snd}\:\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{{n}} \right){dx} \\ $$

Question Number 124482 Answers: 0 Comments: 0

∫e^(sinx) (((xcos^2 x−sinx)/(cos^2 x)))dx

$$\int\mathrm{e}^{\mathrm{sinx}} \left(\frac{\mathrm{xcos}^{\mathrm{2}} \mathrm{x}−\mathrm{sinx}}{\mathrm{cos}^{\mathrm{2}} \mathrm{x}}\right)\mathrm{dx} \\ $$

Question Number 124452 Answers: 1 Comments: 1

2 ((2x+1))^(1/3) = x^3 −1

$$\:\mathrm{2}\:\sqrt[{\mathrm{3}}]{\mathrm{2}{x}+\mathrm{1}}\:=\:{x}^{\mathrm{3}} −\mathrm{1}\: \\ $$

Question Number 124438 Answers: 1 Comments: 2

∫e^((xsinx+cosx)) ∙(((x^4 cos^3 x−xsinx+cosx)/(x^2 cos^2 x)))dx

$$\int\mathrm{e}^{\left(\mathrm{xsinx}+\mathrm{cosx}\right)} \centerdot\left(\frac{\mathrm{x}^{\mathrm{4}} \mathrm{cos}^{\mathrm{3}} \mathrm{x}−\mathrm{xsinx}+\mathrm{cosx}}{\mathrm{x}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \mathrm{x}}\right)\mathrm{dx} \\ $$

Question Number 124432 Answers: 3 Comments: 0

... nice calculus... find:: φ=∫_0 ^( 4) ((ln(x))/((4x−x^2 )^(1/2) ))dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:{find}:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\mathrm{4}} \frac{{ln}\left({x}\right)}{\left(\mathrm{4}{x}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} }{dx}=? \\ $$

Question Number 124430 Answers: 1 Comments: 0

∫ _0 ^( 5) (x^2 /(x^2 +(5−x)^2 )) dx =?

$$\:\int\overset{\:\mathrm{5}} {\:}_{\mathrm{0}} \frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} +\left(\mathrm{5}−{x}\right)^{\mathrm{2}} }\:{dx}\:=?\: \\ $$

Question Number 124421 Answers: 0 Comments: 3

Prove that ∫e^x ∙((x^4 +2)/((1+x^2 )^(5/2) ))dx=((e^x {1+x^2 +x})/((1+x^2 )^(3/2) ))+C

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\int\mathrm{e}^{\mathrm{x}} \centerdot\frac{\mathrm{x}^{\mathrm{4}} +\mathrm{2}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{5}/\mathrm{2}} }\mathrm{dx}=\frac{\mathrm{e}^{\mathrm{x}} \left\{\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right\}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} }+\mathrm{C} \\ $$

Question Number 124371 Answers: 2 Comments: 1

∫ ((e^x (2−sin 2x))/(1−cos 2x)) dx

$$\:\int\:\frac{{e}^{{x}} \left(\mathrm{2}−\mathrm{sin}\:\mathrm{2}{x}\right)}{\mathrm{1}−\mathrm{cos}\:\mathrm{2}{x}}\:{dx}\: \\ $$

Question Number 124360 Answers: 1 Comments: 0

∫_( 0) ^( ∞) ((ln(x))/(x^4 +x^2 +1))dx

$$\int_{\:\mathrm{0}} ^{\:\infty} \frac{\mathrm{ln}\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\mathrm{dx} \\ $$

Question Number 124352 Answers: 0 Comments: 3

∫_0 ^(4π) ∥cosx∥=?

$$\underset{\mathrm{0}} {\overset{\mathrm{4}\pi} {\int}}\parallel{cosx}\parallel=? \\ $$

Question Number 124334 Answers: 1 Comments: 0

∫(√((cosx−cos^3 x)/((1−cos^3 x))))dx

$$\int\sqrt{\frac{\mathrm{cos}{x}−\mathrm{cos}^{\mathrm{3}} {x}}{\left(\mathrm{1}−\mathrm{cos}^{\mathrm{3}} {x}\right)}}\mathrm{d}{x} \\ $$

Question Number 124261 Answers: 3 Comments: 2

∫((2x^2 +5x+9)/((x+1)(√(x^2 +x+1))))dx

$$\int\frac{\mathrm{2x}^{\mathrm{2}} +\mathrm{5x}+\mathrm{9}}{\left(\mathrm{x}+\mathrm{1}\right)\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}}\mathrm{dx} \\ $$

Question Number 124252 Answers: 1 Comments: 0

o(x)=∫ (dx/(sec^3 x sin^4 x))

$$\:\:{o}\left({x}\right)=\int\:\frac{{dx}}{\mathrm{sec}\:^{\mathrm{3}} {x}\:\mathrm{sin}\:^{\mathrm{4}} {x}}\: \\ $$

Question Number 124251 Answers: 0 Comments: 1

∫_0 ^∞ (x^2 /(cosh x)) dx ?

$$\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{x}^{\mathrm{2}} }{\mathrm{cosh}\:{x}}\:{dx}\:? \\ $$

Question Number 124228 Answers: 2 Comments: 0

...::: nice calculus:::... evaluate I=∫_0 ^( ∞) ((cos(ln(x)))/(1+x^3 ))dx=....???

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...:::\:\:{nice}\:\:{calculus}:::... \\ $$$$\:\:\:\:\:\:{evaluate} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\left({ln}\left({x}\right)\right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx}=....??? \\ $$

Question Number 124217 Answers: 1 Comments: 0

∫_0 ^∞ (dx/(x (√(1+x^2 )))) ?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{dx}}{{x}\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:? \\ $$

Question Number 124202 Answers: 0 Comments: 0

::: nice calculus ::: prove that ::: Σ_(m,n=1) ^∞ {(((−1)^(n+m) )/(n^2 +m^2 ))} =^(???) (π^2 /(12)) −(π/4)ln(2)

$$\:\:\:\:\:\:\:\:\:\::::\:\:{nice}\:\:{calculus}\:::: \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:::: \\ $$$$ \\ $$$$\underset{{m},{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{\frac{\left(−\mathrm{1}\right)^{{n}+{m}} }{{n}^{\mathrm{2}} +{m}^{\mathrm{2}} }\right\}\:\overset{???} {=}\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\:−\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{2}\right) \\ $$

Question Number 124201 Answers: 2 Comments: 0

::: nice calculus ::: please prove ::: Ω = ∫_0 ^( ∞) (x^(1/2) /(x^2 +2x+5))dx=(π/( (√ϕ))) where ϕ is Golden ratio...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\::::\:\:{nice}\:\:{calculus}\:::: \\ $$$$\:\:\:\:\:{please}\:\:{prove}\:::: \\ $$$$\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{5}}{dx}=\frac{\pi}{\:\sqrt{\varphi}} \\ $$$$\:\:\:\:\:{where}\:\:\varphi\:\:{is}\:{Golden}\:{ratio}... \\ $$

Question Number 124135 Answers: 1 Comments: 0

B = ∫_0 ^2 [(√((4−x)/x)) − (√(x/(4−x))) ] dx

$$\:{B}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\left[\sqrt{\frac{\mathrm{4}−{x}}{{x}}}\:−\:\sqrt{\frac{{x}}{\mathrm{4}−{x}}}\:\right]\:{dx} \\ $$

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