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Question Number 142438 Answers: 1 Comments: 0

∫_0 ^1 ((1−x)/(lnx))dx how many tricks solve this

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}}{{lnx}}{dx} \\ $$$${how}\:\:{many}\:{tricks}\:{solve}\:{this} \\ $$

Question Number 142437 Answers: 1 Comments: 0

Question Number 142435 Answers: 0 Comments: 1

Question Number 142430 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((log^2 x)/(1+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{log}^{\mathrm{2}} {x}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 142429 Answers: 1 Comments: 0

calculate U_n =∫_0 ^∞ ((log^n x)/(1+x^n ))dx find nature of the serie ΣU_n

$${calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{log}^{{n}} {x}}{\mathrm{1}+{x}^{{n}} }{dx} \\ $$$${find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma{U}_{{n}} \\ $$

Question Number 142426 Answers: 1 Comments: 0

find the value of ∫_0 ^∞ ((xlogx)/((1+x^3 )^2 ))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{xlogx}}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 142425 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((log^3 x)/(1+x^3 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{log}^{\mathrm{3}} {x}}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 142424 Answers: 0 Comments: 0

2)calculate Σ_(k=1) ^(n−1) sin(((kπ)/n)) (n>2) 1) use Rieman sum to prove that ∫_0 ^π log(sinx)dx=−πlog2

$$\left.\mathrm{2}\right){calculate}\:\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{sin}\left(\frac{{k}\pi}{{n}}\right)\:\:\:\left({n}>\mathrm{2}\right) \\ $$$$\left.\mathrm{1}\right)\:{use}\:{Rieman}\:{sum}\:{to}\:{prove} \\ $$$${that}\:\int_{\mathrm{0}} ^{\pi} {log}\left({sinx}\right){dx}=−\pi{log}\mathrm{2} \\ $$

Question Number 142423 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ ((log^2 x)/(1+x^2 ))dx

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{log}^{\mathrm{2}} {x}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 142420 Answers: 0 Comments: 0

Question Number 142415 Answers: 1 Comments: 0

(x/(x+4))=((5⌊x⌋−7)/(7⌊x⌋−5)) x=?

$$\frac{{x}}{{x}+\mathrm{4}}=\frac{\mathrm{5}\lfloor{x}\rfloor−\mathrm{7}}{\mathrm{7}\lfloor{x}\rfloor−\mathrm{5}} \\ $$$${x}=? \\ $$

Question Number 142414 Answers: 1 Comments: 2

4sin^2 18° + 2sin18° + 2,5 = ?

$$\mathrm{4}{sin}^{\mathrm{2}} \mathrm{18}°\:+\:\mathrm{2}{sin}\mathrm{18}°\:+\:\mathrm{2},\mathrm{5}\:=\:? \\ $$

Question Number 142408 Answers: 1 Comments: 1

Question Number 142404 Answers: 2 Comments: 1

Question Number 142401 Answers: 1 Comments: 1

Question Number 142395 Answers: 0 Comments: 0

Question Number 142397 Answers: 0 Comments: 0

Question Number 142393 Answers: 1 Comments: 0

Σ_(k=0) ^(n−1) sec^2 (((kπ)/n))=n^2 ......???

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\mathrm{sec}^{\mathrm{2}} \left(\frac{\mathrm{k}\pi}{\mathrm{n}}\right)=\mathrm{n}^{\mathrm{2}} ......??? \\ $$

Question Number 142389 Answers: 1 Comments: 0

∫(e^x /(cosx))dx

$$\int\frac{{e}^{{x}} }{{cosx}}{dx} \\ $$

Question Number 142388 Answers: 1 Comments: 0

n ∈ N, b, a ∈ N ; a≠0. In base 10; n=aabb^(−) 1. show that n is not prime. 2. Give conditions on b such that n is perfect square. 3. Determinate n such that n is a perfect square.

$$\mathrm{n}\:\in\:\mathbb{N},\:\mathrm{b},\:\mathrm{a}\:\in\:\mathbb{N}\:;\:\mathrm{a}\neq\mathrm{0}. \\ $$$$\mathrm{In}\:\mathrm{base}\:\mathrm{10};\:\mathrm{n}=\overline {\mathrm{aabb}}\: \\ $$$$\mathrm{1}.\:\mathrm{show}\:\mathrm{that}\:\mathrm{n}\:\mathrm{is}\:\mathrm{not}\:\mathrm{prime}. \\ $$$$\mathrm{2}.\:\mathrm{Give}\:\mathrm{conditions}\:\mathrm{on}\:\mathrm{b}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{n}\:\mathrm{is}\:\mathrm{perfect}\:\mathrm{square}. \\ $$$$\mathrm{3}.\:\mathrm{Determinate}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{perfect}\:\mathrm{square}. \\ $$

Question Number 142383 Answers: 1 Comments: 1

Question Number 142379 Answers: 0 Comments: 1

Show that 1+3n<n^2 for every positive integer n≥4

$${Show}\:{that}\:\mathrm{1}+\mathrm{3}{n}<{n}^{\mathrm{2}} \:{for}\:{every}\:{positive}\:{integer}\:{n}\geqslant\mathrm{4} \\ $$

Question Number 142365 Answers: 1 Comments: 0

calculate ∫ (√(1+e^x +e^(2x) ))dx

$$\mathrm{calculate}\:\int\:\:\sqrt{\mathrm{1}+\mathrm{e}^{\mathrm{x}} \:+\mathrm{e}^{\mathrm{2x}} }\mathrm{dx} \\ $$

Question Number 142362 Answers: 1 Comments: 0

prove that: ∫_0 ^( ∞) ln((1/x)).j_0 (x)dx:= γ+ln(2) Hint:(1) j_0 (x)=Σ_(n=0) ^∞ (((−1)^n x^(2n) )/(2^(2n) .Γ^2 (n+1))) (Bessel function) Hint:2 L [ j_0 (x)]=(1/( (√(1+s^2 ))))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\frac{\mathrm{1}}{{x}}\right).{j}_{\mathrm{0}} \left({x}\right){dx}:=\:\gamma+{ln}\left(\mathrm{2}\right)\: \\ $$$$\:\:\:\:\:{Hint}:\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:{j}_{\mathrm{0}} \left({x}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\mathrm{2}^{\mathrm{2}{n}} .\Gamma^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:\left({Bessel}\:{function}\right) \\ $$$$\:\:\:\:{Hint}:\mathrm{2}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathscr{L}\:\left[\:{j}_{\mathrm{0}} \left({x}\right)\right]=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{s}^{\mathrm{2}} }} \\ $$

Question Number 142361 Answers: 1 Comments: 0

Question Number 142359 Answers: 0 Comments: 0

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