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Question Number 166875 Answers: 2 Comments: 1

Question Number 166872 Answers: 2 Comments: 0

∫_0 ^( ∞) (( e^( −x) .ln(x).sin(x))/x) dx = −(π/8) (2γ + ln(2))

$$ \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{e}^{\:−{x}} .{ln}\left({x}\right).{sin}\left({x}\right)}{{x}}\:{dx}\:=\:−\frac{\pi}{\mathrm{8}}\:\left(\mathrm{2}\gamma\:+\:{ln}\left(\mathrm{2}\right)\right) \\ $$$$ \\ $$

Question Number 166861 Answers: 3 Comments: 7

Question Number 166839 Answers: 1 Comments: 0

∫_(−oo) ^(+oo) ((xe^x )/((1+e^x )^2 ))dx

$$\int_{−{oo}} ^{+{oo}} \frac{{xe}^{{x}} }{\left(\mathrm{1}+{e}^{{x}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 166834 Answers: 2 Comments: 0

Question Number 166830 Answers: 1 Comments: 0

Question Number 166829 Answers: 1 Comments: 0

calculate If , f(x)=(( (x^2 +1)(((x^( 2) +x−2)(x^( 4) −1)(x^( 2) +2x−3)+16))^(1/3) + (√(x^( 2) +3)))/(( 1+x +x^( 2) ))) then , f ′ (1 ) =? ■ m.n

$$ \\ $$$$\:\:\:\:\:\:\:\:{calculate}\: \\ $$$$\:\:\:\mathrm{I}{f}\:,\:\:\:\:{f}\left({x}\right)=\frac{\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt[{\mathrm{3}}]{\left({x}^{\:\mathrm{2}} +{x}−\mathrm{2}\right)\left({x}^{\:\mathrm{4}} −\mathrm{1}\right)\left({x}^{\:\mathrm{2}} +\mathrm{2}{x}−\mathrm{3}\right)+\mathrm{16}}\:\:+\:\sqrt{{x}^{\:\mathrm{2}} +\mathrm{3}}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} \right)} \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{then}\:,\:\:\:\:{f}\:'\:\left(\mathrm{1}\:\right)\:=?\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$ \\ $$

Question Number 166828 Answers: 0 Comments: 0

calculate: lim_(n→∞) Σ_(k=1) ^n (1/(n+k^α )) .(0<α<1)

$$\mathrm{calculate}:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}^{\alpha} }\:\:\:\:\:\:\:\:\:\:\:\:.\left(\mathrm{0}<\alpha<\mathrm{1}\right) \\ $$

Question Number 166822 Answers: 1 Comments: 0

Question Number 166818 Answers: 1 Comments: 3

Question Number 166814 Answers: 2 Comments: 0

f(x)=x^n e^(−nx) Determinate f^((n)) (x).

$${f}\left({x}\right)={x}^{{n}} {e}^{−{nx}} \\ $$$${Determinate}\:{f}^{\left({n}\right)} \left({x}\right). \\ $$

Question Number 166813 Answers: 2 Comments: 0

f(x)=x^(2n) Determinate f^((n)) (x)

$${f}\left({x}\right)={x}^{\mathrm{2}{n}} \\ $$$${Determinate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$

Question Number 166812 Answers: 0 Comments: 0

f(x)=e^(nx) cos(x). Determinate f^((n)) (x).

$${f}\left({x}\right)={e}^{{nx}} {cos}\left({x}\right). \\ $$$${Determinate}\:{f}^{\left({n}\right)} \left({x}\right). \\ $$

Question Number 166805 Answers: 1 Comments: 0

f(x)=(x/(1+∣x∣)). show that ∃ K ∈ R_+ such that ∀ x, y ∈ R, ∣f(x)−f(y)∣≤K∣x−y∣

$${f}\left({x}\right)=\frac{{x}}{\mathrm{1}+\mid{x}\mid}. \\ $$$${show}\:{that}\:\exists\:{K}\:\in\:\mathbb{R}_{+} \:{such}\:{that} \\ $$$$\forall\:{x},\:{y}\:\in\:\mathbb{R},\:\mid{f}\left({x}\right)−{f}\left({y}\right)\mid\leqslant{K}\mid{x}−{y}\mid \\ $$

Question Number 166801 Answers: 1 Comments: 2

Calculate: lim_(x→0) ((x−(√x))/( ((sin(x)−tan^2 (x)))^(1/3) ))

$${Calculate}: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{{x}−\sqrt{{x}}}{\:\sqrt[{\mathrm{3}}]{{sin}\left({x}\right)−{tan}^{\mathrm{2}} \left({x}\right)}} \\ $$

Question Number 166888 Answers: 1 Comments: 0

Question Number 166892 Answers: 2 Comments: 0

solve in R i: ⌊ x ⌊ x⌋⌋= 3x ii : ⌊x ⌋^( 2) −3 ⌊x ⌋ +2 ≤ 0 −−−−−−

$$ \\ $$$$\:\:\:{solve}\:{in}\:\:\mathbb{R} \\ $$$$ \\ $$$$\:\:{i}:\:\:\:\:\lfloor\:{x}\:\lfloor\:{x}\rfloor\rfloor=\:\mathrm{3}{x} \\ $$$$ \\ $$$$\:\:{ii}\::\:\:\:\lfloor{x}\:\rfloor^{\:\mathrm{2}} −\mathrm{3}\:\lfloor{x}\:\rfloor\:+\mathrm{2}\:\leqslant\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:−−−−−− \\ $$$$ \\ $$

Question Number 166891 Answers: 0 Comments: 0

Linearize sin^4 (x)cos^2 (x)

$${Linearize}\:{sin}^{\mathrm{4}} \left({x}\right){cos}^{\mathrm{2}} \left({x}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 166796 Answers: 1 Comments: 0

∫ ((sin^3 x)/((cos^2 x+1)(√(cos^2 x+1)))) dx

$$\:\:\:\:\int\:\frac{\mathrm{sin}\:^{\mathrm{3}} \mathrm{x}}{\left(\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}\right)\sqrt{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}}}\:\mathrm{dx} \\ $$

Question Number 166793 Answers: 1 Comments: 0

Question Number 166797 Answers: 0 Comments: 0

∫((√(x+(√x)))/( (√(x+1)))) dx

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

Question Number 166787 Answers: 0 Comments: 3

How many permutations of the letters of the word EINSTEIN are possible if the EIN groups must not be next to eachother?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{permutations}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\mathrm{EINSTEIN}\:\mathrm{are} \\ $$$$\mathrm{possible}\:\mathrm{if}\:\mathrm{the}\:\mathrm{EIN}\:\mathrm{groups}\:\mathrm{must}\:\mathrm{not} \\ $$$$\mathrm{be}\:\mathrm{next}\:\mathrm{to}\:\mathrm{eachother}? \\ $$

Question Number 166779 Answers: 1 Comments: 0

Question Number 166774 Answers: 0 Comments: 1

Question Number 166771 Answers: 0 Comments: 0

lim_(x→0) (((√(1−cos x))−((sin x))^(1/3) )/(sin^2 x)) =?

$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}−\sqrt[{\mathrm{3}}]{\mathrm{sin}\:\mathrm{x}}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}\:=? \\ $$

Question Number 166770 Answers: 4 Comments: 6

In how many ways can you go up a staircase with 20 steps if you take one, two or three steps at a time?

$${In}\:{how}\:{many}\:{ways}\:{can}\:{you}\:{go}\:{up}\:{a}\: \\ $$$${staircase}\:{with}\:\mathrm{20}\:{steps}\:{if}\:{you}\:{take}\: \\ $$$${one},\:{two}\:{or}\:{three}\:{steps}\:{at}\:{a}\:{time}? \\ $$

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