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Question Number 222479    Answers: 4   Comments: 1

Question Number 222478    Answers: 1   Comments: 0

S=Σ_(n=1) ^∞ (−1)^(n−1) (H_n /n^2 ) = ? note: H_n =1+(1/2) +(1/3) +...+(1/n)

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{S}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \frac{{H}_{{n}} }{{n}^{\mathrm{2}} }\:=\:? \\ $$$$\:{note}:\:\:\:{H}_{{n}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{3}}\:+...+\frac{\mathrm{1}}{{n}}\: \\ $$

Question Number 222466    Answers: 0   Comments: 4

find the nth term.

$$\mathrm{find}\:\mathrm{the}\:\mathrm{nth}\:\mathrm{term}.\: \\ $$

Question Number 222462    Answers: 2   Comments: 0

i^i =??

$${i}^{{i}} =?? \\ $$

Question Number 222453    Answers: 1   Comments: 0

∫_0 ^( ∞) ((tanh^2 (x))/x^2 ) dx

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

Question Number 222448    Answers: 0   Comments: 0

∫_0 ^1 ((ln(1+x^2 +(√(x^4 +4x^2 +4))))/(1+x^2 )) dx

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

Question Number 222441    Answers: 2   Comments: 0

0^i

$$\mathrm{0}^{{i}} \\ $$

Question Number 222427    Answers: 1   Comments: 0

if lim_(x→0) (((sin2x)/x^3 )+(a/x^2 )+b)=1 find a and b without using LHopial rule

$$\:\:\boldsymbol{{if}}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\boldsymbol{{sin}}\mathrm{2}\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{3}} }+\frac{\boldsymbol{{a}}}{\boldsymbol{{x}}^{\mathrm{2}} }+\boldsymbol{{b}}\right)=\mathrm{1}\: \\ $$$$\:\:\:\:\boldsymbol{{find}}\:\boldsymbol{{a}}\:\boldsymbol{{and}}\:\boldsymbol{{b}}\:\:\boldsymbol{{without}} \\ $$$$\:\:\:\:\:\:\boldsymbol{{using}}\:\boldsymbol{{LH}}{opial}\:{rule} \\ $$

Question Number 222425    Answers: 3   Comments: 0

lim_(x→∞) (4x+(√(16x^2 −3x))) ans:(3/8)

$$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\infty} \left(\mathrm{4}\boldsymbol{\mathrm{x}}+\sqrt{\mathrm{16}\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{3}\boldsymbol{\mathrm{x}}}\right) \\ $$$$\boldsymbol{\mathrm{ans}}:\frac{\mathrm{3}}{\mathrm{8}} \\ $$

Question Number 222424    Answers: 1   Comments: 0

∫_2 ^( ∞) (dz/(ln(z)))−Σ_(l=2) ^∞ (1/(ln(l)))=??

$$\int_{\mathrm{2}} ^{\:\infty} \:\:\:\:\frac{\mathrm{d}{z}}{\mathrm{ln}\left({z}\right)}−\underset{{l}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{ln}\left({l}\right)}=?? \\ $$

Question Number 222422    Answers: 0   Comments: 0

Prove:∫_0 ^(+∞) ((x^2 lnsinhx)/(cosh 3x))dx=(1/9)π^2 G−(5/(108))π^3 ln 2

$$\mathrm{Prove}:\int_{\mathrm{0}} ^{+\infty} \frac{{x}^{\mathrm{2}} \mathrm{lnsinh}{x}}{\mathrm{cosh}\:\mathrm{3}{x}}{dx}=\frac{\mathrm{1}}{\mathrm{9}}\pi^{\mathrm{2}} {G}−\frac{\mathrm{5}}{\mathrm{108}}\pi^{\mathrm{3}} \mathrm{ln}\:\mathrm{2} \\ $$

Question Number 222419    Answers: 0   Comments: 1

Question Number 222418    Answers: 1   Comments: 0

Prove that: lim_(n→+∞) [ ln^2 (n)−2∫^( n) _( 0) ((lnt)/( (√(1+t^2 )))) dt ]= (π^2 /6)+ln^2 (2)

$$\mathrm{Prove}\:\mathrm{that}:\: \\ $$$$\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\:\left[\:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{n}\right)−\mathrm{2}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{n}} \frac{\mathrm{lnt}}{\:\sqrt{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\:\mathrm{dt}\:\right]=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}}+\mathrm{ln}^{\mathrm{2}} \left(\mathrm{2}\right) \\ $$

Question Number 222415    Answers: 3   Comments: 0

Question Number 222432    Answers: 0   Comments: 0

∫_0 ^(π/2) ((xsinxcosx)/(tan^2 x+cotan^2 x))dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{xsinxcosx}}{{tan}^{\mathrm{2}} {x}+{cotan}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 222411    Answers: 3   Comments: 0

[ 1 .] ∫_0 ^( 1) ((ln(x) ln(1−x^2 )ln(1+x^2 ))/(1−x^2 )) dx [ 2 .] ∫_0 ^1 ((ln(x) ln(1−x) ln(1+x) ln(1+x^2 ))/(1+x)) dx [ 3 .] ∫_0 ^1 ((ln(x) ln(1−x^2 ) ln(1+x^2 ))/x) dx [ 4 .] ∫_0 ^( 1) ((ln(x) ln(1−x) ln(1+x) ln(1−x^2 ))/x) dx

$$\:\: \\ $$$$\:\:\:\left[\:\mathrm{1}\:.\right]\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left({x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}−{x}^{\mathrm{2}} }\:\:\:{dx}\:\:\:\:\: \\ $$$$\:\:\:\left[\:\mathrm{2}\:.\right]\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{ln}\left({x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}\right)\:\mathrm{ln}\left(\mathrm{1}+{x}\right)\:\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}}\:{dx} \\ $$$$\:\left[\:\mathrm{3}\:.\right]\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left({x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\:\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}}\:\:{dx}\:\:\:\: \\ $$$$\:\:\:\left[\:\mathrm{4}\:.\right]\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left({x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}\right)\:\mathrm{ln}\left(\mathrm{1}+{x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}}\:\:{dx}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 222568    Answers: 0   Comments: 0

f(x)=((√(x−2)))^0 and g(x)=(√((x−2)^0 )) dom f(x)=? , dom g(x)=?

$${f}\left({x}\right)=\left(\sqrt{{x}−\mathrm{2}}\right)^{\mathrm{0}} \:\:{and}\:{g}\left({x}\right)=\sqrt{\left({x}−\mathrm{2}\right)^{\mathrm{0}} } \\ $$$${dom}\:{f}\left({x}\right)=?\:,\:{dom}\:{g}\left({x}\right)=? \\ $$

Question Number 222436    Answers: 1   Comments: 0

Question Number 222409    Answers: 0   Comments: 0

Solve ; ∫_0 ^(π/2) ((ln^n sin θ)/(sin^p θ cos^q θ)) dθ , for n,p,q ∈ R_(≥ 0)

$$ \\ $$$$\:\:\:\mathrm{Solve}\:;\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{ln}^{{n}} \:\mathrm{sin}\:\theta}{\mathrm{sin}^{{p}} \:\theta\:\mathrm{cos}^{{q}} \:\theta}\:\mathrm{d}\theta\:,\:\mathrm{for}\:{n},{p},{q}\:\in\:\mathbb{R}_{\geqslant\:\mathrm{0}} \:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 222404    Answers: 1   Comments: 0

Question Number 222408    Answers: 1   Comments: 0

∫_1 ^∞ ((1/x))^(x/( (√(x−1)))) dx = ??

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

Question Number 222389    Answers: 0   Comments: 0

Prove: ∫_0 ^∞ ((cos(nx)cos(p arctan x))/((1+x^2 )^(p/2) ))=(π/2) ((n^(p−1) e^(−n) )/(Γ(p))) (p>0)

$$\mathrm{Prove}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\left({nx}\right)\mathrm{cos}\left({p}\:\mathrm{arctan}\:{x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{p}}{\mathrm{2}}} }=\frac{\pi}{\mathrm{2}}\:\frac{{n}^{{p}−\mathrm{1}} {e}^{−{n}} }{\Gamma\left({p}\right)}\:\left({p}>\mathrm{0}\right) \\ $$

Question Number 222385    Answers: 1   Comments: 2

Question Number 222376    Answers: 1   Comments: 0

Question Number 222373    Answers: 2   Comments: 0

An 80 kg man floats with 4% of his volume above the surface in fresh water.what is his volume? what volume would be above the surface in Sea water?How great is the upthrust on him in air due to the air he displaces? density of sea water =1030kgm^-3, density of fresh water=1000kgm^-3?

An 80 kg man floats with 4% of his volume above the surface in fresh water.what is his volume? what volume would be above the surface in Sea water?How great is the upthrust on him in air due to the air he displaces? density of sea water =1030kgm^-3, density of fresh water=1000kgm^-3?

Question Number 222356    Answers: 1   Comments: 0

∫_0 ^( ∞) f(r)dr=1 , ∫_0 ^( ∞) g(r)dr=1 ∫_(−∞i+𝛄) ^( ∞i+𝛄) F(t)G(t)dt=?? F(t)=∫_0 ^( ∞) f(r)e^(−rt) dr , G(t)=∫_0 ^( ∞) g(r)e^(−rt) dr

$$\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({r}\right)\mathrm{d}{r}=\mathrm{1}\:,\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{g}\left({r}\right)\mathrm{d}{r}=\mathrm{1} \\ $$$$\int_{−\infty\boldsymbol{{i}}+\boldsymbol{\gamma}} ^{\:\:\infty\boldsymbol{{i}}+\boldsymbol{\gamma}} \:{F}\left({t}\right){G}\left({t}\right)\mathrm{d}{t}=?? \\ $$$${F}\left({t}\right)=\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({r}\right){e}^{−{rt}} \mathrm{d}{r}\:,\:{G}\left({t}\right)=\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{g}\left({r}\right){e}^{−{rt}} \mathrm{d}{r} \\ $$

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