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Question Number 73396 Answers: 1 Comments: 1

find ∫_0 ^1 e^(−t^2 ) ln(1−t)dt

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−{t}^{\mathrm{2}} } {ln}\left(\mathrm{1}−{t}\right){dt} \\ $$

Question Number 73378 Answers: 0 Comments: 3

Hello ,i shar withe you nice problem show that ∀k∈N^∗ ∃n∈N such that k≤Σ_(j=1) ^n (1/j)<k+1 have a very Nice day

$${Hello}\:,{i}\:{shar}\:{withe}\:{you}\:{nice}\:{problem}\: \\ $$$${show}\:{that}\:\forall{k}\in\mathbb{N}^{\ast} \:\exists{n}\in\mathbb{N}\:{such}\:{that} \\ $$$${k}\leqslant\underset{{j}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{j}}<{k}+\mathrm{1} \\ $$$${have}\:{a}\:{very}\:{Nice}\:{day} \\ $$$$ \\ $$

Question Number 73358 Answers: 0 Comments: 1

1/4x2−1/2x−13=0

$$\mathrm{1}/\mathrm{4}{x}\mathrm{2}−\mathrm{1}/\mathrm{2}{x}−\mathrm{13}=\mathrm{0} \\ $$

Question Number 73356 Answers: 0 Comments: 2

Question Number 73347 Answers: 0 Comments: 0

Question Number 73346 Answers: 3 Comments: 5

Question Number 73340 Answers: 0 Comments: 2

Is there any pi/product notation rules I discovered some such as: Π_(k=a) ^b [k]=((b!)/((a−1)!)) Π_(k=a) ^b [c]=c^(b−a+1) Π_(k=a) ^b [c∙k]=Π_(k=a) ^b [c]∙Π_(k=a) ^b [k] Π_(k=a) ^b [k+c]=Π_(k=a+c) ^(b+c) [k] But what about this Π_(k=a) ^b [c∙k+d]

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{pi}/\mathrm{product}\:\mathrm{notation}\:\mathrm{rules} \\ $$$$\mathrm{I}\:\mathrm{discovered}\:\mathrm{some}\:\mathrm{such}\:\mathrm{as}: \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{k}\right]=\frac{{b}!}{\left({a}−\mathrm{1}\right)!} \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{c}\right]={c}^{{b}−{a}+\mathrm{1}} \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{c}\centerdot{k}\right]=\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{c}\right]\centerdot\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{k}\right] \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{k}+{c}\right]=\underset{{k}={a}+{c}} {\overset{{b}+{c}} {\prod}}\left[{k}\right] \\ $$$$\mathrm{But}\:\mathrm{what}\:\mathrm{about}\:\mathrm{this} \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{c}\centerdot{k}+{d}\right] \\ $$

Question Number 73338 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((arctan(2cosx))/(3+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left(\mathrm{2}{cosx}\right)}{\mathrm{3}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 73337 Answers: 0 Comments: 3

calculate ∫_0 ^∞ ((cos(artan(2x)))/((3+x^2 )^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({artan}\left(\mathrm{2}{x}\right)\right)}{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 73336 Answers: 1 Comments: 1

find ∫_0 ^∞ e^(−t) ln(1+e^t )dt

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}} {ln}\left(\mathrm{1}+{e}^{{t}} \right){dt} \\ $$

Question Number 73335 Answers: 1 Comments: 2

eplcit f(x)=∫_0 ^1 ln(x+t+t^2 )dt with x>(1/4) 2)calculate ∫_0 ^1 ln(t^2 +t +(√2))dt

$${eplcit}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}+{t}+{t}^{\mathrm{2}} \right){dt}\:\:\:\:\:\:{with}\:{x}>\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({t}^{\mathrm{2}} \:+{t}\:+\sqrt{\mathrm{2}}\right){dt} \\ $$

Question Number 73334 Answers: 1 Comments: 0

calculate lim_(n→+∞) n^2 ( e^(sin((π/n^2 ))) −cos((π/n)))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:{n}^{\mathrm{2}} \left(\:{e}^{{sin}\left(\frac{\pi}{{n}^{\mathrm{2}} }\right)} −{cos}\left(\frac{\pi}{{n}}\right)\right) \\ $$

Question Number 73333 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(π +2x^2 ))/((x^2 +4)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi\:+\mathrm{2}{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 73332 Answers: 0 Comments: 0

let U_n =Σ_(k=0) ^n (((−1)^k )/(√(2k+1))) determine a equivalent of n when n→+∞

$${let}\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{\mathrm{2}{k}+\mathrm{1}}}\:\:{determine}\:{a}\:{equivalent}\:{of}\:{n}\:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 73331 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((ln(1+e^(−3x^2 ) ))/(3+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{x}^{\mathrm{2}} } \right)}{\mathrm{3}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 73330 Answers: 1 Comments: 1

find lim_(x→0) ((ln(2−cos(2x)))/(ln(1+xsin(3x))))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left(\mathrm{2}−{cos}\left(\mathrm{2}{x}\right)\right)}{{ln}\left(\mathrm{1}+{xsin}\left(\mathrm{3}{x}\right)\right)} \\ $$

Question Number 73327 Answers: 0 Comments: 1

let w(x)=∫_0 ^∞ ((lnt)/((x^2 +t^2 )^2 ))dt 1) explicit w(x) 2) calculate U_n =∫_0 ^∞ ((lnt)/((n^2 +t^2 )^2 ))dt find lim_(n→+∞) n^4 U_n and determine nature of tbe serie Σ U_n

$${let}\:{w}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{explicit}\:{w}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left({n}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{4}} {U}_{{n}} \:\:{and}\:{determine}\:{nature}\:{of}\:{tbe}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 73308 Answers: 0 Comments: 6

what are the solutions of (√(3x^2 +1))=n where n∈N

$${what}\:{are}\:{the}\:{solutions} \\ $$$${of}\:\sqrt{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{1}}={n}\:{where}\:{n}\in\mathbb{N} \\ $$

Question Number 73297 Answers: 2 Comments: 1