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...advanced ∗∗∗∗∗∗∗∗∗∗ calculus... prove that ::: :: 𝛗=∫_0 ^( ∞) ((sin(x^4 )ln(x))/x)dx=−((𝛑𝛄)/(32)) note : ∫_0 ^( ∞) ((sin(x)ln(x))/x)dx=^(why???) ((−𝛑𝛄)/2) 𝛗=^(⟨x^4 =t⟩) (1/4)∫_0 ^( ∞) ((sin(t)ln(t^(1/4) ))/t^(1/4) ) ∗(1/t^(3/4) )dt =(1/(16))∫_0 ^( ∞) ((sin(t)ln(t))/t)dt =^(note) ((−𝛑𝛄)/(32)) ( 𝛗=−((𝛑𝛄)/(32)) ) notice:: you will prove that::∫_0 ^( ∞) ((sin(x)ln(x))/x)dx=((−𝛑𝛄)/2) Hint:: ∫_0 ^( ∞) ((sin(x))/(x^( 𝛍) ))dx=^(???) (𝛑/(2𝚪(𝛍)sin(((𝛍𝛑)/2)))) ... m.n.july.1970 ...

Question Number 131527 Answers: 3 Comments: 0

Question Number 131520 Answers: 0 Comments: 2

Nice integral ∫_0 ^( ∞) (x^2 /((x+100)^3 )) dx =?

$$\:\mathrm{Nice}\:\mathrm{integral}\: \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{x}+\mathrm{100}\right)^{\mathrm{3}} }\:\mathrm{dx}\:=? \\ $$

Question Number 131517 Answers: 1 Comments: 0

Given ∫_0 ^3 f(x)dx=∫_0 ^3 (2x−1)dx+∫_0 ^3 (∫_0 ^3 f(x)dx)dx find ∫_(−1) ^1 f(x)dx .

$$\mathrm{Given}\:\int_{\mathrm{0}} ^{\mathrm{3}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\int_{\mathrm{0}} ^{\mathrm{3}} \left(\mathrm{2x}−\mathrm{1}\right)\mathrm{dx}+\int_{\mathrm{0}} ^{\mathrm{3}} \left(\int_{\mathrm{0}} ^{\mathrm{3}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\right)\mathrm{dx} \\ $$$$\mathrm{find}\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:. \\ $$

Question Number 131505 Answers: 4 Comments: 1

If ∫_a ^b f(x)dx = ∫_a ^b g(x)dx is f(x)=g(x) ? true or false?

$$\mathrm{If}\:\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\int_{\mathrm{a}} ^{\mathrm{b}} \mathrm{g}\left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\mathrm{is}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{g}\left(\mathrm{x}\right)\:?\: \\ $$$$\mathrm{true}\:\mathrm{or}\:\mathrm{false}? \\ $$

Question Number 131496 Answers: 3 Comments: 0

Given f(x)=f(x+(π/6)) ∀x∈R if ∫_0 ^( (π/6)) f(x)dx= T find the value of ∫_π ^( ((4π)/3)) f(x+π)dx. nice integral

$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\frac{\pi}{\mathrm{6}}\right)\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\mathrm{if}\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\:\mathrm{T}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\int_{\pi} ^{\:\frac{\mathrm{4}\pi}{\mathrm{3}}} \mathrm{f}\left(\mathrm{x}+\pi\right)\mathrm{dx}. \\ $$$$\mathrm{nice}\:\mathrm{integral} \\ $$

Question Number 131405 Answers: 5 Comments: 2

…… super cooles Integral ⋰⋰ ∫_0 ^( ∞) (dx/((1+x^2 )^2 )) =?

$$ \\ $$$$\:\:\ldots\ldots\:\:\mathrm{super}\:\mathrm{cooles}\:\mathrm{Integral}\:\iddots\iddots \\ $$$$\:\int_{\mathrm{0}} ^{\:\infty} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:=? \\ $$

Question Number 131401 Answers: 1 Comments: 0

...nice calculus... prove that::: ∫_0 ^( 1) (Arcsin(x)).(Arccos(x))dx=^(??) 2−(π/2)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:{prove}\:\:{that}::: \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathscr{A}{rcsin}\left({x}\right)\right).\left(\mathscr{A}{rccos}\left({x}\right)\right){dx}\overset{??} {=}\mathrm{2}−\frac{\pi}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 131364 Answers: 1 Comments: 0

How can I calculate the volume of a region bounded by y=x^2 +3 ;x=1 and x=2 rotating about the y=7 using the shell method.

$$ \\ $$$$\mathrm{How}\:\mathrm{can}\:\mathrm{I}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{volume}\: \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{region}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} +\mathrm{3}\:;\mathrm{x}=\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{x}=\mathrm{2}\:\mathrm{rotating}\:\mathrm{about}\:\mathrm{the}\:\mathrm{y}=\mathrm{7}\:\mathrm{using} \\ $$$$\mathrm{the}\:\mathrm{shell}\:\mathrm{method}. \\ $$

Question Number 131286 Answers: 2 Comments: 0

... calculus .... find :: i:: Σ_(n=2) ^∞ (((−1)^n )/(n^2 −1))=? ii:: Σ_(n=2) ^∞ ((((−1)^n )/(n^4 −1)))=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{calculus}\:.... \\ $$$$\:\:\:\:\:{find}\:::\:\:{i}::\:\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} −\mathrm{1}}=? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\left(\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{4}} −\mathrm{1}}\right)=? \\ $$$$\:\:\:\: \\ $$$$ \\ $$

Question Number 131247 Answers: 8 Comments: 0

(1) ψ = ∫ (dx/(1−sin x cos x))=? (2) ∫_0 ^( ∞) (x/(x^3 +1)) dx =? (3) ∫_0 ^∞ (1/(x^(3/2) +1)) dx =?

$$\left(\mathrm{1}\right)\:\psi\:=\:\int\:\frac{{dx}}{\mathrm{1}−\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}=? \\ $$$$\left(\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{x}}{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}\:=? \\ $$$$\left(\mathrm{3}\right)\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{x}^{\mathrm{3}/\mathrm{2}} +\mathrm{1}}\:{dx}\:=?\: \\ $$

Question Number 131245 Answers: 2 Comments: 0

Find ∫_(−b) ^b ∫_(−a) ^a ((dxdy)/( (x^2 +y^2 +h^2 )^(3/2) ))

$$\:\mathrm{Find}\: \\ $$$$\:\:\:\:\int_{−\mathrm{b}} ^{\mathrm{b}} \int_{−\mathrm{a}} ^{\mathrm{a}} \frac{\mathrm{d}{xdy}}{\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{h}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} } \\ $$

Question Number 131227 Answers: 1 Comments: 0

... real analysis ... prove:: 𝛀= ∫_0 ^( 1) ln(ln((1/x)))ln^2 (x)dx=3−2γ

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{real}\:\:{analysis}\:... \\ $$$$\:\:\:\:\:\:{prove}:: \\ $$$$\:\:\:\boldsymbol{\Omega}=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left({ln}\left(\frac{\mathrm{1}}{{x}}\right)\right){ln}^{\mathrm{2}} \left({x}\right){dx}=\mathrm{3}−\mathrm{2}\gamma \\ $$$$ \\ $$

Question Number 131211 Answers: 1 Comments: 0

...calculus... prove that:: 𝚽=∫_0 ^( (𝛑/4)) (((√(tan(x)+tan^2 (x)))/( (√(tan(x)−tan^2 (x))))) sin(x))dx =(1/2)+((√𝛑)/8) (((𝚪((1/4)))/(𝚪((3/4))))−((𝚪((3/4)))/(𝚪((5/4)))))

$$\:\:\:\:\:\:\:\:\:\:\:\:...{calculus}... \\ $$$$\:{prove}\:{that}:: \\ $$$$\:\boldsymbol{\Phi}=\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} \left(\frac{\sqrt{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)+\boldsymbol{{tan}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)}}{\:\sqrt{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)−\boldsymbol{{tan}}^{\mathrm{2}} \left(\boldsymbol{{x}}\right)}}\:\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\right)\boldsymbol{{dx}}\: \\ $$$$\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\boldsymbol{\pi}}}{\mathrm{8}}\:\left(\frac{\boldsymbol{\Gamma}\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\boldsymbol{\Gamma}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}−\frac{\boldsymbol{\Gamma}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\boldsymbol{\Gamma}\left(\frac{\mathrm{5}}{\mathrm{4}}\right)}\right) \\ $$

Question Number 131189 Answers: 0 Comments: 0

∫((−sin(ln(k^x )))/( (√k)))dx=??

$$\int\frac{−{sin}\left({ln}\left({k}^{{x}} \right)\right)}{\:\sqrt{{k}}}{dx}=?? \\ $$

Question Number 131187 Answers: 0 Comments: 0

Question Number 131183 Answers: 1 Comments: 0

Question Number 131176 Answers: 1 Comments: 0

∫((cos(ln(a^x )))/( (√k)))dx=??

$$\int\frac{{cos}\left({ln}\left({a}^{{x}} \right)\right)}{\:\sqrt{{k}}}{dx}=?? \\ $$

Question Number 131167 Answers: 2 Comments: 0

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