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Question Number 218397 Answers: 0 Comments: 0

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Question Number 218393 Answers: 1 Comments: 0

let ABC be a triangle with incenter I. prove that Ia . Ib . Ic ≤ ((abc)/8)

$$ \\ $$$$\:{let}\:{ABC}\:{be}\:{a}\:{triangle}\:{with}\:{incenter}\:{I}. \\ $$$$\:{prove}\:{that}\:{Ia}\:.\:{Ib}\:.\:{Ic}\:\:\leqslant\:\frac{{abc}}{\mathrm{8}}\:\: \\ $$$$ \\ $$

Question Number 218383 Answers: 0 Comments: 2

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∫_0 ^∞ ((J_𝛂 (ar))/((r^2 +k^2 )𝛍))dr

$$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{J}}_{\boldsymbol{\alpha}} \left(\boldsymbol{{ar}}\right)}{\left(\boldsymbol{{r}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} \right)\boldsymbol{\mu}}\boldsymbol{{dr}}\: \\ $$$$ \\ $$

Question Number 218375 Answers: 1 Comments: 0

∫((√(tan x))/(sin^3 x cos x))dx

$$ \\ $$$$\:\:\int\frac{\sqrt{\boldsymbol{{tan}}\:\boldsymbol{{x}}}}{\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{x}}\:\boldsymbol{{cos}}\:\boldsymbol{{x}}}\boldsymbol{{dx}} \\ $$$$ \\ $$

Question Number 218374 Answers: 0 Comments: 0

∫_0 ^∞ J_α ((√(ar)))e^(−r) dr

$$\: \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \boldsymbol{{J}}_{\alpha} \left(\sqrt{\boldsymbol{{ar}}}\right)\boldsymbol{{e}}^{−\boldsymbol{{r}}} \boldsymbol{{dr}}\: \\ $$$$\: \\ $$

Question Number 218366 Answers: 0 Comments: 2

Question Number 218365 Answers: 1 Comments: 0

Find: 𝛀 =lim_(n→∞) Σ_(k=1) ^n [ Σ_(i=1) ^k i (k − i + (1/3))]^(−1) = ?

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\left[\:\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\:\mathrm{i}\:\left(\mathrm{k}\:−\:\mathrm{i}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\right)\right]^{−\mathrm{1}} =\:? \\ $$

Question Number 218364 Answers: 1 Comments: 0

Let 𝛌>0 fixed Solve for real numbers the system: { ((x^2 − yz = λ^2 )),((y^2 − zx = 7λ^2 )),((z^2 − xy = −5λ^2 )) :}

$$\mathrm{Let}\:\:\:\boldsymbol{\lambda}>\mathrm{0}\:\:\:\mathrm{fixed} \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{system}:\:\:\:\begin{cases}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{yz}\:=\:\lambda^{\mathrm{2}} }\\{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{zx}\:=\:\mathrm{7}\lambda^{\mathrm{2}} }\\{\mathrm{z}^{\mathrm{2}} \:−\:\mathrm{xy}\:=\:−\mathrm{5}\lambda^{\mathrm{2}} }\end{cases} \\ $$

Question Number 218358 Answers: 0 Comments: 0

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Question Number 218331 Answers: 1 Comments: 0

∫_0 ^∞ ((x^2 sin (x))/(1+x^4 )) dx = ?

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} \:\mathrm{sin}\:\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:=\:? \\ $$

Question Number 218328 Answers: 1 Comments: 0

Question Number 218310 Answers: 2 Comments: 1

10 couples are invited to a dinner and should be seated at a round table. in how many ways can the host do this, 1) generally. 2) if the husband and the wife of each couple should sit together. 3) if no two men should sit next to each other. 4) as 3), but two speicial couples should not sit next to each other. it means that the husband from couple 1 may not sit next to the wife from couple 2 and the husband from couple 2 may not sit next to the wife from couple 1. but certainly the husbands of both couples may sit next to their own wifes.

$$\mathrm{10}\:{couples}\:{are}\:{invited}\:{to}\:{a}\:{dinner} \\ $$$${and}\:{should}\:{be}\:{seated}\:{at}\:{a}\:{round}\:{table}. \\ $$$${in}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{host}\:{do} \\ $$$${this}, \\ $$$$\left.\mathrm{1}\right)\:{generally}. \\ $$$$\left.\mathrm{2}\right)\:{if}\:{the}\:{husband}\:{and}\:{the}\:{wife}\:{of} \\ $$$$\:\:\:\:\:{each}\:{couple}\:{should}\:{sit}\:{together}. \\ $$$$\left.\mathrm{3}\right)\:{if}\:{no}\:{two}\:{men}\:{should}\:{sit}\:{next} \\ $$$$\:\:\:\:\:{to}\:{each}\:{other}. \\ $$$$\left.\mathrm{4}\left.\right)\:{as}\:\mathrm{3}\right),\:{but}\:{two}\:{speicial}\:{couples}\: \\ $$$$\:\:\:\:\:{should}\:{not}\:{sit}\:{next}\:{to}\:{each}\:{other}. \\ $$$$\:\:\:\:\underline{\:{it}\:{means}\:}{that}\:{the}\:{husband}\:{from}\: \\ $$$$\:\:\:\:\:{couple}\:\mathrm{1}\:{may}\:{not}\:{sit}\:{next}\:{to}\:{the} \\ $$$$\:\:\:\:\:{wife}\:{from}\:{couple}\:\mathrm{2}\:{and}\:{the} \\ $$$$\:\:\:\:\:\:{husband}\:{from}\:{couple}\:\mathrm{2}\:{may}\:{not} \\ $$$$\:\:\:\:\:\:{sit}\:{next}\:{to}\:{the}\:{wife}\:{from}\:{couple}\:\mathrm{1}. \\ $$$$\:\:\:\:\:\:{but}\:{certainly}\:{the}\:{husbands}\:{of} \\ $$$$\:\:\:\:\:\:{both}\:{couples}\:{may}\:{sit}\:{next}\:{to}\:{their} \\ $$$$\:\:\:\:\:\:{own}\:{wifes}. \\ $$

Question Number 218322 Answers: 1 Comments: 0

Evaluate ∫_0 ^(π/2) ((sin(x))/(sin^3 (x)+cos^3 (x))) dx.

$$\mathrm{Evaluate}\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{\mathrm{sin}\left({x}\right)}{\mathrm{sin}^{\mathrm{3}} \left({x}\right)+\mathrm{cos}^{\mathrm{3}} \left({x}\right)}\:{dx}. \\ $$

Question Number 218318 Answers: 1 Comments: 3

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