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Question Number 167310 Answers: 0 Comments: 0

NICE CALCULUS ∫_0 ^1 ((ln(1−x)ln(1−(x/2)))/x)dx=?

$$\boldsymbol{\mathrm{NICE}}\:\boldsymbol{\mathrm{CALCULUS}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\right)}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}}=? \\ $$$$ \\ $$$$ \\ $$

Question Number 167309 Answers: 0 Comments: 0

Question Number 167305 Answers: 1 Comments: 0

Q=∫ ((2sin (x))/( (√3) sin (x)−cos (x))) dx=?

$$\:\:\:{Q}=\int\:\frac{\mathrm{2sin}\:\left({x}\right)}{\:\sqrt{\mathrm{3}}\:\mathrm{sin}\:\left({x}\right)−\mathrm{cos}\:\left({x}\right)}\:{dx}=? \\ $$

Question Number 167301 Answers: 3 Comments: 0

Question Number 167296 Answers: 1 Comments: 3

if ∣ x−3∣<2 then prove that ∣x^2 −9∣≤ 16

$$\boldsymbol{{if}}\:\mid\:\boldsymbol{{x}}−\mathrm{3}\mid<\mathrm{2}\:\boldsymbol{{then}}\:\boldsymbol{{prove}}\:\boldsymbol{{that}}\:\mid\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{9}\mid\leqslant\:\mathrm{16} \\ $$

Question Number 167295 Answers: 2 Comments: 0

Question Number 170532 Answers: 1 Comments: 0

Let I_n =∫x^n e^(−x) dx, n = 0,1,2,... (i) Show that I_n = −x^n e^(−x) +nI_(n−1) (ii) Show that ∫_0 ^∞ x^n e^(−x) dx = n!

$$\mathrm{Let}\:{I}_{{n}} \:=\int{x}^{{n}} {e}^{−{x}} {dx},\:{n}\:=\:\mathrm{0},\mathrm{1},\mathrm{2},... \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Show}\:\mathrm{that}\:{I}_{{n}} \:=\:−{x}^{{n}} {e}^{−{x}} +{nI}_{{n}−\mathrm{1}} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} {x}^{{n}} {e}^{−{x}} {dx}\:=\:{n}! \\ $$

Question Number 167282 Answers: 2 Comments: 0

solve (3^x )^2 ×4^x =6(√6)

$$\mathrm{solve}\:\left(\mathrm{3}^{\mathrm{x}} \right)^{\mathrm{2}} ×\mathrm{4}^{\mathrm{x}} =\mathrm{6}\sqrt{\mathrm{6}} \\ $$

Question Number 167280 Answers: 1 Comments: 1

Question Number 167289 Answers: 0 Comments: 0

calculate :: lim_(n→∞) (1/n)Σ_(k=1) ^n n^(1/k) =2

$$\mathrm{calculate}\:::\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{n}^{\frac{\mathrm{1}}{\mathrm{k}}} =\mathrm{2} \\ $$

Question Number 167278 Answers: 1 Comments: 0

Question Number 167277 Answers: 1 Comments: 0

lim_(x→0) ((x^3 −3x+3arctan x)/x^5 ) =?

$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{3x}+\mathrm{3arctan}\:\mathrm{x}}{\mathrm{x}^{\mathrm{5}} }\:=? \\ $$

Question Number 167276 Answers: 0 Comments: 0

Question Number 167275 Answers: 0 Comments: 0

Question Number 167268 Answers: 0 Comments: 1

x^(2/( (2)^(1/5) )) =25 x=?

$${x}^{\frac{\mathrm{2}}{\:\sqrt[{\mathrm{5}}]{\mathrm{2}}}} =\mathrm{25}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}=? \\ $$

Question Number 167261 Answers: 1 Comments: 0

Find the probability of obtaining a total of 10 with a throw of two dice if one die has been thrown and shows a 4

$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{of} \\ $$$$\mathrm{obtaining}\:\:\mathrm{a}\:\mathrm{total}\:\mathrm{of}\:\mathrm{10}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{throw}\:\mathrm{of}\:\:\mathrm{two}\:\mathrm{dice}\:\mathrm{if}\:\mathrm{one}\:\mathrm{die} \\ $$$$\mathrm{has}\:\mathrm{been}\:\mathrm{thrown}\:\mathrm{and}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{4} \\ $$

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=?

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

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