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AllQuestion and Answers: Page 508

Question Number 167336 Answers: 1 Comments: 0

0< x<(π/2) f(x)= ((sin(x) ))^(1/(20)) + ((cos(x)))^(1/(20)) R_( f) =? (Range )

$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{0}<\:{x}<\frac{\pi}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:{f}\left({x}\right)=\:\sqrt[{\mathrm{20}}]{{sin}\left({x}\right)\:\:\:}\:+\:\sqrt[{\mathrm{20}}]{{cos}\left({x}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:{R}_{\:{f}} \:=?\:\:\left({Range}\:\right) \\ $$$$ \\ $$

Question Number 167335 Answers: 2 Comments: 3

x_n =n∙sin(2πen!) Calculate lim_(n→∞) x_n

$$\mathrm{x}_{\mathrm{n}} =\mathrm{n}\centerdot\mathrm{sin}\left(\mathrm{2}\pi\mathrm{en}!\right) \\ $$$$\mathrm{Calculate} \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}x}_{\mathrm{n}} \\ $$

Question Number 167334 Answers: 2 Comments: 0

b^b .2^b^b^2 = 4 ⇒b=?

$$\:\:\:\:{b}^{{b}} .\mathrm{2}^{{b}^{{b}^{\mathrm{2}} } } \:=\:\mathrm{4}\:\Rightarrow{b}=? \\ $$

Question Number 167332 Answers: 1 Comments: 0

lim_(n→∞) Σ_(k=1) ^n ((1/k))^k

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\frac{\mathrm{1}}{{k}}\right)^{{k}} \\ $$

Question Number 167330 Answers: 2 Comments: 0

Calculate ∫(1/(x+(√(x^2 +x+1))))dx

$${Calculate} \\ $$$$\int\frac{\mathrm{1}}{{x}+\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$

Question Number 167328 Answers: 1 Comments: 0

∫_0 ^( π/2) ((cos x+cos^5 x sin x)/( (√(1+cos^2 x)))) dx =?

$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \:\frac{\mathrm{cos}\:{x}+\mathrm{cos}\:^{\mathrm{5}} {x}\:\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}}\:{dx}\:=? \\ $$

Question Number 167324 Answers: 2 Comments: 0

Question Number 167321 Answers: 0 Comments: 3

Question Number 167318 Answers: 1 Comments: 0

∫ (dx/( (√(x+(x)^(1/3) )))) =?

$$\:\:\:\:\:\:\int\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}+\sqrt[{\mathrm{3}}]{\mathrm{x}}}}\:=? \\ $$

Question Number 167312 Answers: 2 Comments: 1

∫ ((sin^3 x+1)/(sin^2 x−1)) dx =?

$$\:\:\:\:\:\:\:\int\:\frac{\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}+\mathrm{1}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{1}}\:\mathrm{dx}\:=? \\ $$

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

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

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