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IntegrationQuestion and Answers: Page 125

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

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

Question Number 124134 Answers: 1 Comments: 0

ϕ(x)= ∫ (((x^3 +2)/x^3 )) (√(x−(1/x^2 ))) dx

$$\:\varphi\left({x}\right)=\:\int\:\left(\frac{{x}^{\mathrm{3}} +\mathrm{2}}{{x}^{\mathrm{3}} }\right)\:\sqrt{{x}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}\:{dx}\: \\ $$

Question Number 124101 Answers: 1 Comments: 0

∫(((x^(−6) −64)/(4+2x^(−1) +x^(−2) ))∙(x^2 /(4−4x^(−1) +x^(−2) ))−((4x^2 (2x+1))/(1−2x)))dx

$$\int\left(\frac{\mathrm{x}^{−\mathrm{6}} −\mathrm{64}}{\mathrm{4}+\mathrm{2x}^{−\mathrm{1}} +\mathrm{x}^{−\mathrm{2}} }\centerdot\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{4}−\mathrm{4x}^{−\mathrm{1}} +\mathrm{x}^{−\mathrm{2}} }−\frac{\mathrm{4x}^{\mathrm{2}} \left(\mathrm{2x}+\mathrm{1}\right)}{\mathrm{1}−\mathrm{2x}}\right)\mathrm{dx} \\ $$

Question Number 124063 Answers: 0 Comments: 0

find ∫∫_D ((arctan((√(x^2 +y^2 ))))/(x+y))dxdy D={(x,y) / 0≤x≤1 and 1≤y≤2}

$${find}\:\:\int\int_{{D}} \frac{{arctan}\left(\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\right)}{{x}+{y}}{dxdy} \\ $$$${D}=\left\{\left({x},{y}\right)\:/\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{1}\leqslant{y}\leqslant\mathrm{2}\right\} \\ $$

Question Number 124062 Answers: 0 Comments: 0

find ∫_(π/2) ^(π/4) ((sin(nx))/(sin^n (x)))dx (n natural)

$${find}\:\int_{\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{sin}\left({nx}\right)}{{sin}^{{n}} \left({x}\right)}{dx}\:\:\:\left({n}\:{natural}\right) \\ $$

Question Number 124059 Answers: 0 Comments: 0

find ∫_0 ^∞ ((x^3 sin(2x))/((x^2 +x+1)^3 ))dx

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

Question Number 124046 Answers: 1 Comments: 0

φ(α) = ∫ ((6α^2 +30α+2)/(4α^2 +20α+25)) dα

$$\:\:\phi\left(\alpha\right)\:=\:\int\:\frac{\mathrm{6}\alpha^{\mathrm{2}} +\mathrm{30}\alpha+\mathrm{2}}{\mathrm{4}\alpha^{\mathrm{2}} +\mathrm{20}\alpha+\mathrm{25}}\:{d}\alpha\: \\ $$

Question Number 124043 Answers: 2 Comments: 0

∫ ((sin x cos x)/(1−2cos 2x)) dx

$$\:\int\:\frac{\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}{\mathrm{1}−\mathrm{2cos}\:\mathrm{2}{x}}\:{dx}\: \\ $$

Question Number 124061 Answers: 2 Comments: 0

find ∫_0 ^∞ ((xarctanx)/((x^(2 ) +1)^2 ))dx

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

Question Number 124024 Answers: 1 Comments: 0

Question Number 124004 Answers: 1 Comments: 0

∫ (dx/( ((tan x))^(1/3) )) ?

$$\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{\mathrm{tan}\:{x}}}\:?\: \\ $$

Question Number 123998 Answers: 4 Comments: 0

∫_( 0) ^( ∞) (x^2 /((1+x^2 )^2 )) dx

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

Question Number 123977 Answers: 2 Comments: 0

Question Number 123964 Answers: 0 Comments: 4

∫(√(x^3 +ax+b))dx

$$\int\sqrt{{x}^{\mathrm{3}} +{ax}+{b}}{dx} \\ $$

Question Number 123967 Answers: 1 Comments: 0

Question Number 123937 Answers: 2 Comments: 0

...nice calculus... prove that:: ∫_((−π)/4) ^(π/4) (((π−4x)tan(x))/(1−tan(x)))dx=^(???) πln(2)−(π^2 /4)

$$\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:\:\:\:{calculus}... \\ $$$$\:{prove}\:{that}:: \\ $$$$\:\int_{\frac{−\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \frac{\left(\pi−\mathrm{4}{x}\right){tan}\left({x}\right)}{\mathrm{1}−{tan}\left({x}\right)}{dx}\overset{???} {=}\pi{ln}\left(\mathrm{2}\right)−\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$$$ \\ $$

Question Number 123921 Answers: 3 Comments: 0

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

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

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