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Question Number 125501    Answers: 0   Comments: 0

find the value of Σ_(n=2) ^∞ (((−1)^n (2n+1))/(n^4 −1))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\sum_{\mathrm{n}=\mathrm{2}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{2n}+\mathrm{1}\right)}{\mathrm{n}^{\mathrm{4}} −\mathrm{1}} \\ $$

Question Number 125500    Answers: 1   Comments: 0

find relation between ∫ f(x)dx and ∫ f^(−1) (x)dx

$$\mathrm{find}\:\mathrm{relation}\:\mathrm{between}\:\int\:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:\mathrm{and}\:\int\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{x}\right)\mathrm{dx}\:\: \\ $$

Question Number 125462    Answers: 1   Comments: 0

... ♣advanced calculus♣... ⧫⧫ prove that: I=∫_1 ^( ∞) (((t^4 −6t^2 +1)ln(ln(t)))/((1+t^2 )^3 ))dt=((2G)/π) G : catalan constant...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\clubsuit{advanced}\:\:{calculus}\clubsuit... \\ $$$$\:\:\:\blacklozenge\blacklozenge\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\mathrm{I}=\int_{\mathrm{1}} ^{\:\infty} \frac{\left({t}^{\mathrm{4}} −\mathrm{6}{t}^{\mathrm{2}} +\mathrm{1}\right){ln}\left({ln}\left({t}\right)\right)}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} }{dt}=\frac{\mathrm{2G}}{\pi} \\ $$$$\:\:\mathrm{G}\::\:\:{catalan}\:\:{constant}... \\ $$

Question Number 125458    Answers: 2   Comments: 0

...advanced calculus... evaluate ::: Σ_(n=2) ^∞ { ((ζ (2n ))/2^( n) ) } =??

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:{calculus}... \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\left\{\:\frac{\zeta\:\left(\mathrm{2}{n}\:\right)}{\mathrm{2}^{\:{n}} }\:\right\}\:=?? \\ $$$$ \\ $$

Question Number 125440    Answers: 2   Comments: 0

∫((2x^2 −3x−3)/((x−1)(x^2 −2x+5))) dx

$$\:\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}−\mathrm{3}}{\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{5}\right)}\:{dx}\: \\ $$

Question Number 125421    Answers: 2   Comments: 0

∫_0 ^(100) (dx/( (√(x(100−x))))) ?

$$\:\underset{\mathrm{0}} {\overset{\mathrm{100}} {\int}}\:\frac{{dx}}{\:\sqrt{{x}\left(\mathrm{100}−{x}\right)}}\:?\: \\ $$

Question Number 125416    Answers: 2   Comments: 0

∫ (dx/( (√(x(√x)−x^2 )))) ?

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

Question Number 125390    Answers: 2   Comments: 0

nice calculus... evaluate ::::↷ Ω=∫_0 ^( ∞) (((√x) tan^(−1) (x))/(1+x^2 ))dx=???

$$\:\:\:\:\:\:\:\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{evaluate}\:::::\curvearrowright \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \frac{\sqrt{{x}}\:{tan}^{−\mathrm{1}} \left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=??? \\ $$

Question Number 125381    Answers: 0   Comments: 0

F(x) = cos (∫_1 ^x cos (∫_1 ^t sin^3 u du )dy) ((dF(x))/dx) = ?

$$\:{F}\left({x}\right)\:=\:\mathrm{cos}\:\left(\underset{\mathrm{1}} {\overset{{x}} {\int}}\:\mathrm{cos}\:\left(\underset{\mathrm{1}} {\overset{{t}} {\int}}\:\mathrm{sin}\:^{\mathrm{3}} {u}\:{du}\:\right){dy}\right) \\ $$$$\:\frac{{dF}\left({x}\right)}{{dx}}\:=\:?\: \\ $$

Question Number 125346    Answers: 4   Comments: 1

∫ x (√(1−x^4 )) dx??

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

Question Number 125336    Answers: 2   Comments: 0

... nice calculus ... evaluate : φ=∫_(0 ) ^( 1) x^2 ln(x).ln(1−x)dx =?

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

Question Number 125313    Answers: 3   Comments: 0

β(x)=∫ (x^3 /( (√(1−x^2 )))) dx

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

Question Number 125276    Answers: 2   Comments: 0

∫_0 ^1 ((x^9 −1)/(lnx))dx=???

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{x}^{\mathrm{9}} −\mathrm{1}}{{lnx}}{dx}=??? \\ $$

Question Number 125234    Answers: 0   Comments: 1

calculate ∫_0 ^1 (dx/(x+2+(√(x^2 +x+1))))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{dx}}{\mathrm{x}+\mathrm{2}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}} \\ $$

Question Number 125233    Answers: 0   Comments: 0

calculate u_(nm) =∫_0 ^∞ e^(−nx) ln(1+e^(mx) )dx find Σ_(n≥0 and m≥0) u_(nm)

$$\mathrm{calculate}\:\mathrm{u}_{\mathrm{nm}} =\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{nx}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{mx}} \right)\mathrm{dx} \\ $$$$\mathrm{find}\:\sum_{\mathrm{n}\geqslant\mathrm{0}\:\mathrm{and}\:\mathrm{m}\geqslant\mathrm{0}} \:\:\mathrm{u}_{\mathrm{nm}} \\ $$

Question Number 125194    Answers: 1   Comments: 0

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

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

Question Number 125187    Answers: 1   Comments: 0

∫ (dx/( (√(sin^3 x)) (√(cos^5 x)))) ?

$$\:\:\int\:\frac{{dx}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}}\:\sqrt{\mathrm{cos}\:^{\mathrm{5}} {x}}}\:? \\ $$

Question Number 125146    Answers: 3   Comments: 0

1)calculate ∫_0 ^(2π) (dθ/(x^2 −2x cosθ +1)) 2) calculate ∫_0 ^(2π) ((cosθ)/((x^2 −2xcosθ +1)^2 ))dθ

$$\left.\mathrm{1}\right)\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{\mathrm{d}\theta}{\mathrm{x}^{\mathrm{2}} −\mathrm{2x}\:\mathrm{cos}\theta\:+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{\mathrm{cos}\theta}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2xcos}\theta\:+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{d}\theta \\ $$

Question Number 125133    Answers: 3   Comments: 0

... ◂advanced calculus▶... prove that ::: Ω=∫_0 ^( 1) {((cos(log(x))−1)/(log(x)))}dx=((log(2))/2) ...∗adopted from youtube∗... ∗ ∗ youtube solution is not considered ∗ ∗

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\blacktriangleleft{advanced}\:\:\:{calculus}\blacktriangleright... \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\:::: \\ $$$$\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{\frac{{cos}\left({log}\left({x}\right)\right)−\mathrm{1}}{{log}\left({x}\right)}\right\}{dx}=\frac{{log}\left(\mathrm{2}\right)}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:...\ast{adopted}\:{from}\:{youtube}\ast...\:\:\: \\ $$$$\:\ast\:\ast\:{youtube}\:{solution}\:{is}\:{not}\:{considered}\:\ast\:\ast \\ $$$$\:\: \\ $$

Question Number 125125    Answers: 0   Comments: 1

∫_(π/6) ((s^(π/3) inx )/x)dx=?

$$\underset{\frac{\pi}{\mathrm{6}}} {\int}\frac{\overset{\frac{\pi}{\mathrm{3}}} {{s}inx}\:}{{x}}{dx}=? \\ $$

Question Number 125114    Answers: 2   Comments: 1

solve ∫ (dx/((x^3 −1)^2 )) ?

$$\:{solve}\:\int\:\frac{{dx}}{\left({x}^{\mathrm{3}} −\mathrm{1}\right)^{\mathrm{2}} }\:? \\ $$

Question Number 125098    Answers: 1   Comments: 1

...nice calculus ... prove that :: Apery′s constant φ=∫_0 ^( 1) {(4x^2 +4^2 x^2^2 +4^3 x^2^3 +...)((ln^2 (x))/(x(1+x)))}dx =2ζ(3)−1

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}\:... \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:::\:{Apery}'{s}\:{constant} \\ $$$$\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left\{\left(\mathrm{4}{x}^{\mathrm{2}} +\mathrm{4}^{\mathrm{2}} {x}^{\mathrm{2}^{\mathrm{2}} } +\mathrm{4}^{\mathrm{3}} {x}^{\mathrm{2}^{\mathrm{3}} } +...\right)\frac{{ln}^{\mathrm{2}} \left({x}\right)}{{x}\left(\mathrm{1}+{x}\right)}\right\}{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\zeta\left(\mathrm{3}\right)−\mathrm{1} \\ $$

Question Number 125096    Answers: 1   Comments: 0

... nice calculus... suppose :: z =x−iy & (z)^(1/3) =p+iq then find :: A=(((x/p)+(y/q))/(p^2 +q^2 )) =?? note : i=(√(−1))

$$\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:\:{calculus}... \\ $$$$\:\:\:\:{suppose}\:::\:{z}\:={x}−{iy}\:\:\&\:\sqrt[{\mathrm{3}}]{{z}}\:={p}+{iq} \\ $$$$\:\:\:{then}\:\:{find}\:::\:\:\:{A}=\frac{\frac{{x}}{{p}}+\frac{{y}}{{q}}}{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} }\:=?? \\ $$$$\:{note}\::\:{i}=\sqrt{−\mathrm{1}} \\ $$

Question Number 125053    Answers: 2   Comments: 1

∫ (dx/( (√(x^2 +3x−4)))) =?

$$\:\:\int\:\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{4}}}\:=? \\ $$

Question Number 125052    Answers: 0   Comments: 3

A rescue cable attached to a helicopter′s weighs 2 lb/ft. A man 180−lb grabs the end of the rope and his pulled from the ocean into the helicopter. How much work is done in lifting the man if the helicopter is 40 ft above the water ? (a) 8800 lb−ft (b) 1780 lb−ft (c) 7280 lb−ft (d) 10,400 lb−ft

$$\:{A}\:{rescue}\:{cable}\:{attached}\:{to}\:{a}\: \\ $$$${helicopter}'{s}\:{weighs}\:\mathrm{2}\:{lb}/{ft}.\: \\ $$$${A}\:{man}\:\mathrm{180}−{lb}\:{grabs}\:{the}\:{end}\: \\ $$$${of}\:{the}\:{rope}\:{and}\:{his}\:{pulled}\: \\ $$$${from}\:{the}\:{ocean}\:{into}\:{the}\:{helicopter}. \\ $$$${How}\:{much}\:{work}\:{is}\:{done}\:{in}\: \\ $$$${lifting}\:{the}\:{man}\:{if}\:{the}\:{helicopter} \\ $$$${is}\:\mathrm{40}\:{ft}\:{above}\:{the}\:{water}\:? \\ $$$$\left({a}\right)\:\mathrm{8800}\:{lb}−{ft} \\ $$$$\left({b}\right)\:\mathrm{1780}\:{lb}−{ft} \\ $$$$\left({c}\right)\:\mathrm{7280}\:{lb}−{ft} \\ $$$$\left({d}\right)\:\mathrm{10},\mathrm{400}\:{lb}−{ft} \\ $$

Question Number 125050    Answers: 1   Comments: 0

∫_0 ^π (e^(cos x) /(e^(cos x) +e^(−cos x) )) dx =?

$$\:\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{{e}^{\mathrm{cos}\:{x}} }{{e}^{\mathrm{cos}\:{x}} +{e}^{−\mathrm{cos}\:{x}} }\:{dx}\:=?\: \\ $$

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