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Question Number 138748 Answers: 1 Comments: 0

A committee of 3 members is to be formed from 8 members. Find the number of committees that can be formed if two particular club members cannot both be in a committee

$$\mathrm{A}\:\mathrm{committee}\:\mathrm{of}\:\mathrm{3}\:\mathrm{members}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{from}\:\mathrm{8}\:\mathrm{members}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{committees}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{if}\:\mathrm{two}\:\mathrm{particular} \\ $$$$\mathrm{club}\:\mathrm{members}\:\mathrm{cannot}\:\mathrm{both}\:\mathrm{be}\:\mathrm{in}\:\mathrm{a}\:\mathrm{committee} \\ $$

Question Number 138742 Answers: 2 Comments: 0

........ nice .... calculus... evaluate :: Ω=∫_0 ^( 1) ((x^(e^π −1) −x^(e^γ −1) )/(ln((x)^(1/3) )))dx=^? 3(π−γ)

$$\:\:\:\:\:\:\:\:\:\:\:........\:{nice}\:\:\:....\:\:\:{calculus}... \\ $$$$\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{x}^{{e}^{\pi} −\mathrm{1}} −{x}^{{e}^{\gamma} −\mathrm{1}} }{{ln}\left(\sqrt[{\mathrm{3}}]{{x}}\:\right)}{dx}\overset{?} {=}\mathrm{3}\left(\pi−\gamma\right) \\ $$

Question Number 138735 Answers: 2 Comments: 0

Question Number 138734 Answers: 0 Comments: 1

Question Number 138733 Answers: 0 Comments: 0

.....mathematical ....analysis..... suppose f :[a , b]→R is a function and α:[a , b]→^(α↗) R (α is an increasing function on [a , b]) meanwhile α is continuous at y_0 where a≤y_0 ≤b . defining f(x)= { (( 1 x=y_0 )),(( 0 x≠y_0 )) :} prove that : f∈ R (α) .... Hint: f∈R (α) ⇔ ∀ ε>0 ∃ P_ε ; U(P_ε ,f,α)−L(P_ε ,f,α)<ε Reimann criterion ....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:....{analysis}..... \\ $$$$\:\:{suppose}\:\:\:\:{f}\::\left[{a}\:,\:{b}\right]\rightarrow\mathbb{R}\:{is}\:{a}\:{function} \\ $$$$\:\:\:{and}\:\:\:\alpha:\left[{a}\:,\:{b}\right]\overset{\alpha\nearrow} {\rightarrow}\mathbb{R}\:\left(\alpha\:{is}\:{an}\:{increasing}\:{function}\right. \\ $$$$\left.\:{on}\:\left[{a}\:,\:{b}\right]\right)\:\:{meanwhile}\:\alpha\:{is}\:{continuous}\:{at}\:{y}_{\mathrm{0}} \: \\ $$$$\:\:{where}\:\:\:{a}\leqslant{y}_{\mathrm{0}} \leqslant{b}\:\:.\:{defining}\: \\ $$$$\:\:\:{f}\left({x}\right)=\begin{cases}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{x}={y}_{\mathrm{0}} }\\{\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:{x}\neq{y}_{\mathrm{0}} }\end{cases} \\ $$$$\:\:\:\:{prove}\:\:{that}\::\:{f}\in\:\mathscr{R}\:\left(\alpha\right)\:.... \\ $$$$\:\:\:\:{Hint}:\:{f}\in\mathscr{R}\:\left(\alpha\right)\:\Leftrightarrow\:\forall\:\epsilon>\mathrm{0}\:\exists\:{P}_{\epsilon} \:;\:{U}\left({P}_{\epsilon} ,{f},\alpha\right)−{L}\left({P}_{\epsilon} ,{f},\alpha\right)<\epsilon \\ $$$$\:\:\:\:{Reimann}\:\:{criterion}\:.... \\ $$

Question Number 138730 Answers: 0 Comments: 0

Question Number 138728 Answers: 0 Comments: 0

∫_0 ^∞ ((ln(1+cos x))/(1+e^x ))dx=0

$$\int_{\mathrm{0}} ^{\infty} \frac{{ln}\left(\mathrm{1}+\mathrm{cos}\:{x}\right)}{\mathrm{1}+{e}^{{x}} }{dx}=\mathrm{0} \\ $$

Question Number 138723 Answers: 0 Comments: 2

........advanced... ... ...math...... prove that _∗^∗ :::: 𝛀=Σ_(k=0) ^∞ {(1/(16^k ))((4/(8k+1))−(2/(8k+4))−(1/(8k+5))−(1/(8k+6)))}=π ....Bailey−Borwein formula....

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:........{advanced}...\:...\:...{math}...... \\ $$$$\:{prove}\:{that}\:_{\ast} ^{\ast} \:\::::: \\ $$$$\:\:\:\boldsymbol{\Omega}=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\mathrm{1}}{\mathrm{16}^{{k}} }\left(\frac{\mathrm{4}}{\mathrm{8}{k}+\mathrm{1}}−\frac{\mathrm{2}}{\mathrm{8}{k}+\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{8}{k}+\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{8}{k}+\mathrm{6}}\right)\right\}=\pi \\ $$$$\:\:\:\:\:\:\:\:\:....{Bailey}−{Borwein}\:{formula}.... \\ $$$$\:\:\: \\ $$

Question Number 138725 Answers: 1 Comments: 0

Solve for real numbers ((sin(sinx))/(sinx)) + ((cos(cosx))/(cosx)) = 1

$${Solve}\:{for}\:{real}\:{numbers} \\ $$$$\frac{{sin}\left({sinx}\right)}{{sinx}}\:+\:\frac{{cos}\left({cosx}\right)}{{cosx}}\:=\:\mathrm{1} \\ $$

Question Number 138716 Answers: 1 Comments: 0

∫_2 ^∞ (1/(x^2 lnx))dx converges or diverges?

$$\int_{\mathrm{2}} ^{\infty} \frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \mathrm{lnx}}\mathrm{dx}\: \\ $$$$\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges}? \\ $$

Question Number 138818 Answers: 0 Comments: 0