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Question Number 186327    Answers: 0   Comments: 0

lim_(x→0) (((2/3)x^4 +x^2 −tan^2 x)/x^6 ) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{2}}{\mathrm{3}}{x}^{\mathrm{4}} +{x}^{\mathrm{2}} −\mathrm{tan}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{6}} }\:=? \\ $$

Question Number 186323    Answers: 1   Comments: 1

Question Number 186322    Answers: 1   Comments: 0

Question Number 186321    Answers: 3   Comments: 0

Question Number 186320    Answers: 1   Comments: 0

Question Number 186310    Answers: 1   Comments: 0

((∫x(x^2 +5)^(1/2) dx − 3∫x(x^2 +5)^(−1/2) dx)/(∫ ((x[(x^2 +5)−3])/( (√(x^2 +5 )))) dx)) =??

$$ \\ $$$$\:\:\:\frac{\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{\mathrm{1}/\mathrm{2}} \boldsymbol{{dx}}\:−\:\mathrm{3}\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{−\mathrm{1}/\mathrm{2}} \:\boldsymbol{{dx}}}{\int\:\:\frac{\boldsymbol{{x}}\left[\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)−\mathrm{3}\right]}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\:\:}}\:\boldsymbol{{dx}}}\:=??\:\:\:\: \\ $$$$ \\ $$

Question Number 186306    Answers: 0   Comments: 2

Evaluate ∫((ln(sin x))/(ln(tan x)+1)) dx

$$\mathrm{Evaluate}\:\int\frac{\mathrm{ln}\left(\mathrm{sin}\:{x}\right)}{\mathrm{ln}\left(\mathrm{tan}\:{x}\right)+\mathrm{1}}\:{dx} \\ $$

Question Number 186305    Answers: 0   Comments: 3

log (((3.2^ )/(3.1^ ))) find Characteristic?

$$\mathrm{log}\:\left(\frac{\mathrm{3}.\bar {\mathrm{2}}}{\mathrm{3}.\bar {\mathrm{1}}}\right)\:\:\:\:\:\:\:{find}\:\mathrm{Characteristic}? \\ $$

Question Number 186302    Answers: 1   Comments: 3

Question Number 186301    Answers: 1   Comments: 0

∫5x+6^x dx

$$\int\mathrm{5}{x}+\mathrm{6}^{{x}} {dx} \\ $$

Question Number 186300    Answers: 1   Comments: 0

solve in R ⌊ 2log_( 8) (x) + (1/3) ⌋ = log_( 4) (x )+ (1/2)

$$\:\:\: \\ $$$$\:\:\:\:\:\mathrm{solve}\:\:\mathrm{in}\:\:\:\mathbb{R} \\ $$$$ \\ $$$$\:\:\:\lfloor\:\:\mathrm{2log}_{\:\mathrm{8}} \left({x}\right)\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\rfloor\:=\:\mathrm{log}_{\:\mathrm{4}} \left({x}\:\right)+\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 186298    Answers: 1   Comments: 0

Question Number 186597    Answers: 1   Comments: 0

Question Number 186594    Answers: 5   Comments: 1

Question Number 186293    Answers: 0   Comments: 4

Question Number 186292    Answers: 0   Comments: 0

(x−p)(x^3 −x−(1/3))=0 x^4 −px^3 −x^2 +(p−(1/3))x+(p/3)=0 (x^2 +ax+h)(x^2 +bx+k)=0 a+b=−p h+k+ab=−1 bh+ak=p−(1/3) hk=(p/3) −−−−−− say ab=t −−−−−− ah+bk+p−(1/3)=p(1+t) bh+ak−(p−(1/3))=0 ⇒ (a−b)(h−k)=pt−p+(2/3) squaring (p^2 −4t){(1+t)^2 −((4p)/3)}=(pt−p+(2/3))^2 say t+1=z (p^2 +4−4z)(z^2 −((4p)/3))=(pz−2p+(2/3))^2 ⇒ −4z^3 +(p^2 +4)z^2 +((16pz)/3)−((4p)/3)(p^2 +4) =p^2 z^2 −4p(p−(1/3))z+4(p−(1/3))^2 ⇒ z^3 −z^2 −p(p+1)z+(p−(1/3))^2 +(p/3)(p^2 +4)=0 .....

$$\left({x}−{p}\right)\left({x}^{\mathrm{3}} −{x}−\frac{\mathrm{1}}{\mathrm{3}}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{4}} −{px}^{\mathrm{3}} −{x}^{\mathrm{2}} +\left({p}−\frac{\mathrm{1}}{\mathrm{3}}\right){x}+\frac{{p}}{\mathrm{3}}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} +{ax}+{h}\right)\left({x}^{\mathrm{2}} +{bx}+{k}\right)=\mathrm{0} \\ $$$${a}+{b}=−{p} \\ $$$${h}+{k}+{ab}=−\mathrm{1} \\ $$$${bh}+{ak}={p}−\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${hk}=\frac{{p}}{\mathrm{3}} \\ $$$$−−−−−− \\ $$$${say}\:{ab}={t} \\ $$$$−−−−−− \\ $$$${ah}+{bk}+{p}−\frac{\mathrm{1}}{\mathrm{3}}={p}\left(\mathrm{1}+{t}\right) \\ $$$${bh}+{ak}−\left({p}−\frac{\mathrm{1}}{\mathrm{3}}\right)=\mathrm{0} \\ $$$$\Rightarrow\:\left({a}−{b}\right)\left({h}−{k}\right)={pt}−{p}+\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${squaring} \\ $$$$\left({p}^{\mathrm{2}} −\mathrm{4}{t}\right)\left\{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} −\frac{\mathrm{4}{p}}{\mathrm{3}}\right\}=\left({pt}−{p}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2}} \\ $$$${say}\:\:{t}+\mathrm{1}={z} \\ $$$$\left({p}^{\mathrm{2}} +\mathrm{4}−\mathrm{4}{z}\right)\left({z}^{\mathrm{2}} −\frac{\mathrm{4}{p}}{\mathrm{3}}\right)=\left({pz}−\mathrm{2}{p}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{2}} \\ $$$$\Rightarrow \\ $$$$−\mathrm{4}{z}^{\mathrm{3}} +\left({p}^{\mathrm{2}} +\mathrm{4}\right){z}^{\mathrm{2}} +\frac{\mathrm{16}{pz}}{\mathrm{3}}−\frac{\mathrm{4}{p}}{\mathrm{3}}\left({p}^{\mathrm{2}} +\mathrm{4}\right) \\ $$$$\:\:\:\:={p}^{\mathrm{2}} {z}^{\mathrm{2}} −\mathrm{4}{p}\left({p}−\frac{\mathrm{1}}{\mathrm{3}}\right){z}+\mathrm{4}\left({p}−\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\:{z}^{\mathrm{3}} −{z}^{\mathrm{2}} −{p}\left({p}+\mathrm{1}\right){z}+\left({p}−\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:+\frac{{p}}{\mathrm{3}}\left({p}^{\mathrm{2}} +\mathrm{4}\right)=\mathrm{0} \\ $$$$.....\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 186285    Answers: 0   Comments: 2

Prove that: R (m , n) ≤ C_(m+n) ^m Here R states the Ramsey theory

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{R}\:\left(\mathrm{m}\:,\:\mathrm{n}\right)\:\leqslant\:\mathrm{C}_{\boldsymbol{\mathrm{m}}+\boldsymbol{\mathrm{n}}} ^{\boldsymbol{\mathrm{m}}} \\ $$$$\mathrm{Here}\:\:\mathrm{R}\:\:\mathrm{states}\:\mathrm{the}\:\:\mathrm{Ramsey}\:\:\mathrm{theory} \\ $$

Question Number 186283    Answers: 0   Comments: 0

Question Number 186282    Answers: 1   Comments: 0

Question Number 186271    Answers: 2   Comments: 0

Question Number 186267    Answers: 1   Comments: 1

lim_(x→0) (((1+2016x)^(2017) −(1+2017x)^(2016) )/x^2 ) without L′H or series

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\mathrm{2016}{x}\right)^{\mathrm{2017}} −\left(\mathrm{1}+\mathrm{2017}{x}\right)^{\mathrm{2016}} }{{x}^{\mathrm{2}} } \\ $$$${without}\:{L}'{H}\:{or}\:{series} \\ $$

Question Number 186265    Answers: 1   Comments: 0

f(x)= (√( x −a)) + (√(3a −x)) with ( a>0) is given .If , f_( max) . f_( min) = (√(32)) find , ” a ” = ?

$$ \\ $$$$\:\:\:{f}\left({x}\right)=\:\sqrt{\:{x}\:−{a}}\:\:+\:\sqrt{\mathrm{3}{a}\:−{x}}\:\:\:{with}\:\left(\:{a}>\mathrm{0}\right) \\ $$$$\:\:\:\:{is}\:\:{given}\:.{If}\:\:,\:\:{f}_{\:{max}} \:.\:{f}_{\:{min}} \:=\:\sqrt{\mathrm{32}} \\ $$$$\:\:\:\:\:{find}\:\:,\:\:\:\:\:\:\:''\:\:\:{a}\:\:''\:\:=\:? \\ $$

Question Number 186263    Answers: 2   Comments: 0

∫_0 ^( 1) (√(1+(1/(4x)))) dx

$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}{x}}}\:{dx} \\ $$$$ \\ $$

Question Number 186256    Answers: 0   Comments: 1

Prove that: R (m , n) ≤ C_(m+n) ^m

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathbb{R}\:\left(\mathrm{m}\:,\:\mathrm{n}\right)\:\leqslant\:\mathbb{C}_{\boldsymbol{\mathrm{m}}+\boldsymbol{\mathrm{n}}} ^{\boldsymbol{\mathrm{m}}} \\ $$$$ \\ $$

Question Number 186253    Answers: 1   Comments: 1

Question Number 186249    Answers: 0   Comments: 0

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