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Question Number 167252 Answers: 1 Comments: 0

Question Number 167248 Answers: 2 Comments: 1

(√3^x )+1=2^x faind x=?

$$\sqrt{\mathrm{3}^{{x}} }+\mathrm{1}=\mathrm{2}^{{x}} \:\:\:\:{faind}\:{x}=? \\ $$

Question Number 167267 Answers: 1 Comments: 0

Question Number 167243 Answers: 1 Comments: 0

log _3 (x^2 −2)< log _3 ((3/2)∣x∣−1)

$$\:\:\:\:\:\:\mathrm{log}\:_{\mathrm{3}} \left({x}^{\mathrm{2}} −\mathrm{2}\right)<\:\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{3}}{\mathrm{2}}\mid{x}\mid−\mathrm{1}\right)\: \\ $$

Question Number 167241 Answers: 2 Comments: 0

(b/(11)) is General fraction and 0.3a^(−) faind (a−9b)=?

$$\frac{{b}}{\mathrm{11}}\:\:{is}\:{General}\:{fraction}\:\:{and}\:\mathrm{0}.\overline {\mathrm{3}{a}} \\ $$$${faind}\:\:\left({a}−\mathrm{9}{b}\right)=? \\ $$

Question Number 167233 Answers: 2 Comments: 0

∫ ((lnx)/(x+1))dx

$$\int\:\frac{{lnx}}{{x}+\mathrm{1}}{dx} \\ $$

Question Number 167231 Answers: 2 Comments: 1

Question Number 167230 Answers: 0 Comments: 0

lim_( α→∞) { (α ∫_0 ^( ∞) sin( x^( α) ) dx )=ϕ(α)]= (π/2) −−−− ∫_0 ^( ∞) sin(x^( α) )dx =^(x^( α) = y) (1/α)∫_0 ^( ∞) (( sin(y))/y^( 1−(1/α)) ) dy ⇒ α ∫_0 ^( ∞) sin(x^( α) ) dx = ∫_0 ^( ∞) (( sin(y))/y^( 1−(1/α)) ) dy = (( π)/(2 Γ (1−(1/α))sin ((π/2) (1−(1/α))))) = (π/(2Γ (1−(1/α))cos ((π/(2α)) ))) = ϕ (α ) lim_( α→∞) ϕ (α )=^((1/α) =β) lim_( β→0) (π/(2Γ (1−β)cos ((π/2) β))) = (π/2)

$$ \\ $$$$\:\:\:\:\:{lim}_{\:\alpha\rightarrow\infty} \left\{\:\left(\alpha\:\int_{\mathrm{0}} ^{\:\infty} {sin}\left(\:{x}^{\:\alpha} \right)\:{dx}\:\right)=\varphi\left(\alpha\right)\right]=\:\frac{\pi}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:−−−− \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {sin}\left({x}^{\:\alpha} \right){dx}\:\overset{{x}^{\:\alpha} =\:{y}} {=}\:\frac{\mathrm{1}}{\alpha}\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left({y}\right)}{{y}^{\:\mathrm{1}−\frac{\mathrm{1}}{\alpha}} }\:{dy} \\ $$$$\:\:\:\:\:\Rightarrow\:\:\alpha\:\int_{\mathrm{0}} ^{\:\infty} {sin}\left({x}^{\:\alpha} \right)\:{dx}\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left({y}\right)}{{y}^{\:\mathrm{1}−\frac{\mathrm{1}}{\alpha}} }\:{dy} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\:\pi}{\mathrm{2}\:\Gamma\:\left(\mathrm{1}−\frac{\mathrm{1}}{\alpha}\right){sin}\:\left(\frac{\pi}{\mathrm{2}}\:\left(\mathrm{1}−\frac{\mathrm{1}}{\alpha}\right)\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\pi}{\mathrm{2}\Gamma\:\left(\mathrm{1}−\frac{\mathrm{1}}{\alpha}\right){cos}\:\left(\frac{\pi}{\mathrm{2}\alpha}\:\right)}\:=\:\varphi\:\left(\alpha\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{lim}_{\:\alpha\rightarrow\infty} \:\varphi\:\left(\alpha\:\right)\overset{\frac{\mathrm{1}}{\alpha}\:=\beta} {=}{lim}_{\:\beta\rightarrow\mathrm{0}} \:\:\frac{\pi}{\mathrm{2}\Gamma\:\left(\mathrm{1}−\beta\right){cos}\:\left(\frac{\pi}{\mathrm{2}}\:\beta\right)}\:=\:\frac{\pi}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 167225 Answers: 0 Comments: 0

Question Number 167223 Answers: 0 Comments: 3

∫ sin^3 (3x) cos^4 (5x) dx=?

$$\:\:\int\:\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{3x}\right)\:\mathrm{cos}\:^{\mathrm{4}} \left(\mathrm{5x}\right)\:\mathrm{dx}=? \\ $$

Question Number 167221 Answers: 1 Comments: 0

The plane y=1 slices the surface z=arctan(((x+y)/(1−xy))) in a curve C. Find the slope of the tangent line to C at x=2

$${The}\:{plane}\:{y}=\mathrm{1}\:{slices}\:{the}\:{surface}\: \\ $$$${z}={arctan}\left(\frac{{x}+{y}}{\mathrm{1}−{xy}}\right) \\ $$$${in}\:{a}\:{curve}\:{C}. \\ $$$${Find}\:{the}\:{slope}\:{of}\:{the}\:{tangent}\:{line}\:{to} \\ $$$${C}\:{at}\:{x}=\mathrm{2} \\ $$

Question Number 167220 Answers: 2 Comments: 0

Question Number 167216 Answers: 0 Comments: 0

Transform the equation 7x−10y+13=0 into: a. Slope intercept form: b. Symmetric form: c. Normal form:

$${Transform}\:{the}\:{equation}\:\mathrm{7}{x}−\mathrm{10}{y}+\mathrm{13}=\mathrm{0} \\ $$$${into}: \\ $$$${a}.\:\:\mathrm{S}{lope}\:{intercept}\:{form}: \\ $$$${b}.\:\:{Symmetric}\:{form}: \\ $$$${c}.\:\:{Normal}\:{form}: \\ $$

Question Number 167215 Answers: 2 Comments: 0

Question Number 167213 Answers: 1 Comments: 0

solve I= ∫_0 ^( (1/2)) ((ln^( 2) (x))/(1−x)) dx =?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{solve} \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\mathcal{I}=\:\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{{ln}^{\:\mathrm{2}} \left({x}\right)}{\mathrm{1}−{x}}\:{dx}\:=? \\ $$$$ \\ $$

Question Number 167202 Answers: 2 Comments: 0

Question Number 167200 Answers: 1 Comments: 0

Question Number 167196 Answers: 0 Comments: 4

# Question # If , a ∉ Z and the function with the following rule is differentiable at ” x = 1 ” then find the value of “ a “ . f(x) = (⌊ (x/2) ⌋ + ⌊ ((−x)/2) ⌋ )∣ x^( 2) +x −2∣ ⌊ax ⌋ ■ m.n

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:#\:{Question}\:# \\ $$$$\:\:\:\:\:{If}\:,\:{a}\:\notin\:\mathbb{Z}\:\:{and}\:\:{the}\:{function}\:{with}\:{the}\:\:{following}\:\:{rule}\:\: \\ $$$$\:\:\:\:\:\:{is}\:\:{differentiable}\:{at}\:\:\:\:''\:\:{x}\:=\:\mathrm{1}\:''\:\:{then}\:\:{find}\:\:{the}\:{value}\:{of} \\ $$$$\:\:\:\:\:\:\:``\:\:\:{a}\:\:``\:\:\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)\:=\:\left(\lfloor\:\frac{{x}}{\mathrm{2}}\:\rfloor\:+\:\lfloor\:\frac{−{x}}{\mathrm{2}}\:\rfloor\:\right)\mid\:{x}^{\:\mathrm{2}} +{x}\:−\mathrm{2}\mid\:\lfloor{ax}\:\rfloor\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$ \\ $$

Question Number 167187 Answers: 0 Comments: 0

Question Number 167185 Answers: 2 Comments: 2

Find the equation of the locus of a moving point P, such that its distance from the point A(4,3) and B(5,1) is in the ratio 3 : 1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{a}\:\mathrm{moving}\:\mathrm{point}\:\mathrm{P}, \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{its}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{point} \\ $$$$\mathrm{A}\left(\mathrm{4},\mathrm{3}\right)\:\mathrm{and}\:\mathrm{B}\left(\mathrm{5},\mathrm{1}\right)\:\mathrm{is}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{3}\::\:\mathrm{1} \\ $$

Question Number 167183 Answers: 5 Comments: 1

Question Number 167181 Answers: 1 Comments: 0

Given x, c ∈ R. (u_n )_(n∈N ) : { ((u_0 =0)),((u_(n+1) =xsin(u_n )+c)) :} 1.Show that ∣u_(n+1) −u_n ∣≤∣c∣∣x^n ∣. 2.Show that : (∣x∣<1 et m≥n ⇒∣u_m −u_n ∣≤((∣c∣∣x^n ∣)/(1−∣x∣)) 3.Deduct that u_n is convergent and calculate its limit.

$${Given}\:{x},\:{c}\:\in\:\mathbb{R}. \\ $$$$\left({u}_{{n}} \right)_{{n}\in\mathbb{N}\:} :\begin{cases}{{u}_{\mathrm{0}} =\mathrm{0}}\\{{u}_{{n}+\mathrm{1}} ={xsin}\left({u}_{{n}} \right)+{c}}\end{cases} \\ $$$$\mathrm{1}.{Show}\:{that}\:\mid{u}_{{n}+\mathrm{1}} −{u}_{{n}} \mid\leqslant\mid{c}\mid\mid{x}^{{n}} \mid. \\ $$$$\mathrm{2}.{Show}\:{that}\::\:\left(\mid{x}\mid<\mathrm{1}\:{et}\:{m}\geqslant{n}\:\Rightarrow\mid{u}_{{m}} −{u}_{{n}} \mid\leqslant\frac{\mid{c}\mid\mid{x}^{{n}} \mid}{\mathrm{1}−\mid{x}\mid}\right. \\ $$$$\mathrm{3}.{Deduct}\:{that}\:{u}_{{n}} \:{is}\:{convergent}\:{and} \\ $$$${calculate}\:{its}\:{limit}. \\ $$

Question Number 167189 Answers: 0 Comments: 1

Question Number 167179 Answers: 1 Comments: 0

Demonstrate that ∀ x, y ∈ R_+ ^∗ , ∀ q ∈ Q_+ ^∗ such that q>xy, ∃ a,b ∈ Q such that a>x, b>y and ab=q.

$${Demonstrate}\:{that}\:\forall\:{x},\:{y}\:\in\:\mathbb{R}_{+} ^{\ast} ,\:\forall\:{q}\:\in\:\mathbb{Q}_{+} ^{\ast} \\ $$$${such}\:{that}\:{q}>{xy},\:\exists\:{a},{b}\:\in\:\mathbb{Q}\:{such}\: \\ $$$${that}\:{a}>{x},\:{b}>{y}\:{and}\:{ab}={q}. \\ $$

Question Number 167176 Answers: 0 Comments: 1

Question Number 167173 Answers: 1 Comments: 0

∫_(−2) ^(−1) e^(−(t/2)) (√(t+2)) dt = ???

$$\int_{−\mathrm{2}} ^{−\mathrm{1}} {e}^{−\frac{{t}}{\mathrm{2}}} \sqrt{{t}+\mathrm{2}}\:{dt}\:=\:??? \\ $$

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