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Question Number 94753    Answers: 1   Comments: 0

Question Number 94730    Answers: 2   Comments: 0

Question Number 94723    Answers: 2   Comments: 0

lim_(x→0) ((∫_0 ^x^2 (√(4+t^3 )) dt)/x^2 ) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{\mathrm{0}} {\overset{{x}^{\mathrm{2}} } {\int}}\:\sqrt{\mathrm{4}+{t}^{\mathrm{3}} \:}\:{dt}}{{x}^{\mathrm{2}} }\:?\: \\ $$

Question Number 94718    Answers: 0   Comments: 4

∫ (√(tan x)) dx = ∫(((√(tan x))+(√(cot x)))/2) dx + ∫ (((√(tan x))−(√(cot x)))/2) dx =(1/((√2) ))∫ ((sin x+cos x)/(√(sin 2x))) dx + (1/((√2) ))∫ ((sin x−cos x)/(√(sin 2x))) dx = (1/((√2) ))∫ ((sin x+cos x)/(√(1−(sin x−cos x)^2 ))) dx + (1/(√2)) ∫ ((sin x−cos x)/(√((sin x+cos x)^2 −1))) dx = (1/((√2) ))∫ (dt/(√(1−t^2 ))) +(1/((√2) ))∫ ((−du)/(√(u^2 −1))) = (1/((√2) ))sin^(−1) (t) −(1/(√2)) ln(u+(√(u^2 −1))) +c = (1/(√2)) sin^(−1) (sin x−cos x)− (1/(√2)) ln (sin x+cos x+(√(sin 2x))) + c where t = sin x−cos x ; u = sin x+cos x

$$\int\:\sqrt{\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}\:= \\ $$$$\int\frac{\sqrt{\mathrm{tan}\:\mathrm{x}}+\sqrt{\mathrm{cot}\:\mathrm{x}}}{\mathrm{2}}\:\mathrm{dx}\:+\:\int\:\frac{\sqrt{\mathrm{tan}\:\mathrm{x}}−\sqrt{\mathrm{cot}\:\mathrm{x}}}{\mathrm{2}}\:\mathrm{dx} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\int\:\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx}\:+\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx} \\ $$$$=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\int\:\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\sqrt{\mathrm{1}−\left(\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} }}\:\mathrm{dx}\:+\: \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\sqrt{\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} −\mathrm{1}}}\:\mathrm{dx}\: \\ $$$$=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\int\:\frac{\mathrm{dt}}{\sqrt{\mathrm{1}−\mathrm{t}^{\mathrm{2}} }}\:+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\int\:\frac{−\mathrm{du}}{\sqrt{\mathrm{u}^{\mathrm{2}} −\mathrm{1}}} \\ $$$$=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{t}\right)\:−\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:\mathrm{ln}\left(\mathrm{u}+\sqrt{\mathrm{u}^{\mathrm{2}} −\mathrm{1}}\right)\:+\mathrm{c} \\ $$$$=\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\right)− \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:\mathrm{ln}\:\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}+\sqrt{\mathrm{sin}\:\mathrm{2x}}\right)\:+\:\mathrm{c}\: \\ $$$$\mathrm{where}\:\mathrm{t}\:=\:\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\:;\: \\ $$$$\mathrm{u}\:=\:\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\: \\ $$

Question Number 94710    Answers: 1   Comments: 2

Question Number 94708    Answers: 1   Comments: 1

Question Number 94706    Answers: 2   Comments: 0

Question Number 94705    Answers: 0   Comments: 1

a set X had one more subset than set Y. If X has 8 more subsets than Y. Find the number if element in the set X.

$$\mathrm{a}\:\mathrm{set}\:\mathrm{X}\:\mathrm{had}\:\mathrm{one}\:\mathrm{more}\:\mathrm{subset}\:\mathrm{than}\:\mathrm{set}\:\mathrm{Y}. \\ $$$$\mathrm{If}\:\mathrm{X}\:\mathrm{has}\:\mathrm{8}\:\mathrm{more}\:\mathrm{subsets}\:\mathrm{than}\:\mathrm{Y}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{if}\:\mathrm{element}\:\mathrm{in}\:\mathrm{the}\:\mathrm{set}\:\mathrm{X}. \\ $$

Question Number 94701    Answers: 0   Comments: 3

what is the volume of x^2 +y^2 +z^2 =2

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{2} \\ $$$$ \\ $$

Question Number 94698    Answers: 1   Comments: 0

Question Number 94693    Answers: 0   Comments: 1

Question Number 94690    Answers: 0   Comments: 4

Question Number 94679    Answers: 2   Comments: 2

Question Number 94676    Answers: 0   Comments: 0

Find the supremum and infimum of A = { n ∣ 3 + (2/n) , n ∈ R }

$${Find}\:\:{the}\:\:{supremum}\:\:{and}\:\:{infimum}\:\:{of}\:\: \\ $$$${A}\:\:=\:\:\left\{\:{n}\:\mid\:\:\mathrm{3}\:+\:\frac{\mathrm{2}}{{n}}\:\:,\:\:{n}\:\in\:\:\mathbb{R}\:\:\right\} \\ $$

Question Number 94664    Answers: 0   Comments: 9

App updates: • Improvement in handling image upload and display (small width, large height cases) • Editor can be opened in landscape mode. • Added ability to save thread to device. ′Save to Device′ • An automatic comment is added to a post to visit referenced question Q/q followed by number ex. 94644 on same line

$$\mathrm{App}\:\mathrm{updates}: \\ $$$$\bullet\:\mathrm{Improvement}\:\mathrm{in}\:\mathrm{handling}\:\mathrm{image} \\ $$$$\:\:\:\:\mathrm{upload}\:\mathrm{and}\:\mathrm{display}\:\left(\mathrm{small}\:\mathrm{width},\right. \\ $$$$\left.\:\:\:\:\mathrm{large}\:\mathrm{height}\:\mathrm{cases}\right) \\ $$$$\bullet\:\mathrm{Editor}\:\mathrm{can}\:\mathrm{be}\:\mathrm{opened}\:\mathrm{in}\:\mathrm{landscape} \\ $$$$\:\:\:\:\mathrm{mode}. \\ $$$$\bullet\:\mathrm{Added}\:\mathrm{ability}\:\mathrm{to}\:\mathrm{save}\:\mathrm{thread}\:\mathrm{to} \\ $$$$\:\:\:\:\mathrm{device}.\:'\mathrm{Save}\:\mathrm{to}\:\mathrm{Device}' \\ $$$$\bullet\:\mathrm{An}\:\mathrm{automatic}\:\mathrm{comment}\:\mathrm{is}\:\mathrm{added} \\ $$$$\:\:\:\:\mathrm{to}\:\mathrm{a}\:\mathrm{post}\:\mathrm{to}\:\mathrm{visit}\:\mathrm{referenced}\:\mathrm{question} \\ $$$$\:\:\:\:\:\mathrm{Q}/\mathrm{q}\:\mathrm{followed}\:\mathrm{by}\:\mathrm{number}\:\mathrm{ex}.\:\mathrm{94644} \\ $$$$\:\:\:\:\:\mathrm{on}\:\mathrm{same}\:\mathrm{line} \\ $$

Question Number 94662    Answers: 1   Comments: 3

∫ (√(tan x+cot x)) dx = ?

$$\int\:\sqrt{\mathrm{tan}\:\mathrm{x}+\mathrm{cot}\:\mathrm{x}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 94661    Answers: 0   Comments: 0

calculate Σ a_n x^n if a_n verify a_(n+1) =a_n +a_(n−1)

$${calculate}\:\Sigma\:{a}_{{n}} {x}^{{n}} \:{if}\:{a}_{{n}} \:{verify} \\ $$$${a}_{{n}+\mathrm{1}} ={a}_{{n}} \:+{a}_{{n}−\mathrm{1}} \\ $$

Question Number 94660    Answers: 0   Comments: 0

u_n =(1+(1/n^2 ))(1+(2/n^2 ))...(1+(n/n^2 )) find lim_(n→+∞) u_n

$${u}_{{n}} =\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\left(\mathrm{1}+\frac{\mathrm{2}}{{n}^{\mathrm{2}} }\right)...\left(\mathrm{1}+\frac{{n}}{{n}^{\mathrm{2}} }\right) \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \\ $$

Question Number 94659    Answers: 1   Comments: 0

calculate lim_(n→+∞) ∫_0 ^∞ (1−(t/n))^n e^(−3t) dt

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \int_{\mathrm{0}} ^{\infty} \:\left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} \:{e}^{−\mathrm{3}{t}} \:{dt} \\ $$

Question Number 94658    Answers: 0   Comments: 0

find lim_(n→+∞) ∫_0 ^1 (1−(t/n))^n arctan(1+nt)dt

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} \:{arctan}\left(\mathrm{1}+{nt}\right){dt} \\ $$

Question Number 94657    Answers: 0   Comments: 0

let u_n =Σ_(k=1) ^n (1/((^p (√k)))) detetmine a equivalent of u_n (n→+∞)

$${let}\:{u}_{{n}} \:=\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{\left(^{{p}} \sqrt{{k}}\right)} \\ $$$${detetmine}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \\ $$$$\left({n}\rightarrow+\infty\right) \\ $$$$ \\ $$

Question Number 94655    Answers: 0   Comments: 0

solve (1+nz)^p +(1−nz)^p =0 n and p ontegr natural

$${solve}\:\left(\mathrm{1}+{nz}\right)^{{p}} +\left(\mathrm{1}−{nz}\right)^{{p}} \:=\mathrm{0} \\ $$$${n}\:{and}\:{p}\:{ontegr}\:{natural} \\ $$

Question Number 94654    Answers: 0   Comments: 0

prove that Σ_(i=1) ^n x_i y_i ≤(Σ_(i=1) ^n x_i ^2 )^(1/2) (Σ_(i=1) ^n y_i ^2 )^(1/2) x_i and y_i reals ≥0

$${prove}\:{that}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{x}_{{i}} {y}_{{i}} \leqslant\left(\sum_{{i}=\mathrm{1}} ^{{n}} {x}_{{i}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\sum_{{i}=\mathrm{1}} ^{{n}} {y}_{{i}} ^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$${x}_{{i}} \:{and}\:{y}_{{i}} \:{reals}\:\geqslant\mathrm{0} \\ $$

Question Number 94652    Answers: 0   Comments: 0

find lim_(ξ→0) ∫_0 ^1 ((sin(ξx))/((√(1+ξx^2 ))−(√(1−ξx^2 ))))dx

$${find}\:{lim}_{\xi\rightarrow\mathrm{0}} \:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{sin}\left(\xi{x}\right)}{\sqrt{\mathrm{1}+\xi{x}^{\mathrm{2}} }−\sqrt{\mathrm{1}−\xi{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 94650    Answers: 1   Comments: 1

let f(x) =(x+1)^9 e^(−3x) calculstr f^((7)) (0) and f^((5)) (1)

$${let}\:{f}\left({x}\right)\:=\left({x}+\mathrm{1}\right)^{\mathrm{9}} \:{e}^{−\mathrm{3}{x}} \\ $$$${calculstr}\:{f}^{\left(\mathrm{7}\right)} \left(\mathrm{0}\right)\:{and}\:{f}^{\left(\mathrm{5}\right)} \left(\mathrm{1}\right) \\ $$

Question Number 94649    Answers: 0   Comments: 0

calculate ∫_0 ^∞ e^(−x) lnx dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {lnx}\:{dx} \\ $$

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