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Question Number 159526 Answers: 0 Comments: 0

Ω= ∫_0 ^( ∞) (( sin^( 3) (x)ln(x))/x) dx=^(??) (π/8) (ln(3)−2γ) −−−−− solution.. Ω=∫_0^ ^( ∞) {(((3/4) sin(x)−(1/4) sin(3x))/x)} ln(x)dx = (3/4) (((−πγ)/2))− (1/4){ ∫_0 ^( ∞) ((sin(3x)ln(x))/x)dx=Ψ} ∴ Ψ := ∫_0 ^( ∞) (( sin(x).[ln(x)−ln(3)])/x)dx := −((πγ)/2) − ((ln(3).π)/2) ∴ Ω := ((−3πγ)/8) +((πγ)/8) +((π.ln(3))/8) := ((−2πγ)/8) + ((π.ln(3))/8) = (π/8) ( ln(3)−2γ )

$$ \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{3}} \left({x}\right){ln}\left({x}\right)}{{x}}\:{dx}\overset{??} {=}\frac{\pi}{\mathrm{8}}\:\left({ln}\left(\mathrm{3}\right)−\mathrm{2}\gamma\right) \\ $$$$−−−−− \\ $$$$\:\:\:\:\:\:{solution}.. \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}^{\:} } ^{\:\infty} \left\{\frac{\frac{\mathrm{3}}{\mathrm{4}}\:{sin}\left({x}\right)−\frac{\mathrm{1}}{\mathrm{4}}\:{sin}\left(\mathrm{3}{x}\right)}{{x}}\right\}\:{ln}\left({x}\right){dx} \\ $$$$\:\:=\:\:\frac{\mathrm{3}}{\mathrm{4}}\:\left(\frac{−\pi\gamma}{\mathrm{2}}\right)−\:\frac{\mathrm{1}}{\mathrm{4}}\left\{\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{3}{x}\right){ln}\left({x}\right)}{{x}}{dx}=\Psi\right\} \\ $$$$\:\:\therefore\:\:\Psi\::=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}\left({x}\right).\left[{ln}\left({x}\right)−{ln}\left(\mathrm{3}\right)\right]}{{x}}{dx} \\ $$$$\:\:\:\:\:\::=\:−\frac{\pi\gamma}{\mathrm{2}}\:\:−\:\frac{{ln}\left(\mathrm{3}\right).\pi}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\:\:\:\:\Omega\::=\:\:\frac{−\mathrm{3}\pi\gamma}{\mathrm{8}}\:+\frac{\pi\gamma}{\mathrm{8}}\:\:+\frac{\pi.{ln}\left(\mathrm{3}\right)}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\::=\:\frac{−\mathrm{2}\pi\gamma}{\mathrm{8}}\:+\:\frac{\pi.{ln}\left(\mathrm{3}\right)}{\mathrm{8}}\:=\:\frac{\pi}{\mathrm{8}}\:\left(\:{ln}\left(\mathrm{3}\right)−\mathrm{2}\gamma\:\right) \\ $$$$ \\ $$

Question Number 159525 Answers: 0 Comments: 0

I:=∫_0 ^( ∞) ((( sin^( 3) (x))/x^( 3) )) ln(x)dx=?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}:=\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\:{sin}^{\:\mathrm{3}} \left({x}\right)}{{x}^{\:\mathrm{3}} }\right)\:{ln}\left({x}\right){dx}=? \\ $$$$ \\ $$

Question Number 159522 Answers: 0 Comments: 1

lim_(x→0) ((sin(sinx))/x)=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\left({sinx}\right)}{{x}}=? \\ $$

Question Number 159517 Answers: 1 Comments: 1

lim_(x→0) (1/x^2 )(e^(−2x) −((1+ax)/(1+bx))) a+b=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\left({e}^{−\mathrm{2}{x}} −\frac{\mathrm{1}+{ax}}{\mathrm{1}+{bx}}\right) \\ $$$${a}+{b}=? \\ $$

Question Number 159507 Answers: 2 Comments: 1

Question Number 159497 Answers: 0 Comments: 4

Prove that 1 + (1/4) + (1/9) + …+ (1/n^2 ) < 2 for n ∈ N .

$${Prove}\:\:{that}\:\: \\ $$$$\:\:\:\:\:\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{1}}{\mathrm{9}}\:+\:\ldots+\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:\:<\:\:\mathrm{2} \\ $$$${for}\:\:{n}\:\in\:\:\mathbb{N}\:\:. \\ $$

Question Number 159494 Answers: 1 Comments: 0

Question Number 159493 Answers: 0 Comments: 0

Question Number 159491 Answers: 1 Comments: 0

Question Number 159490 Answers: 0 Comments: 1

Question Number 159488 Answers: 2 Comments: 0

Question Number 159480 Answers: 1 Comments: 1

Question Number 159479 Answers: 1 Comments: 0

Question Number 159477 Answers: 1 Comments: 0

Find: Ω =∫ (1/((x + (1/x))^2 )) dx

$$\mathrm{Find}:\:\:\:\Omega\:=\int\:\frac{\mathrm{1}}{\left(\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 159475 Answers: 0 Comments: 0

Question Number 159474 Answers: 1 Comments: 0

nice integral. prove that ∫^( ∞) _0 (( tan^( −1) (2x)+ tan^( −1) ((x/2) ))/(1+x^( 2) ))dx=(π^( 2) /4)

$$ \\ $$$$\:\:\:\:\:\:{nice}\:\:{integral}. \\ $$$$\:\:{prove}\:\:{that} \\ $$$$\underset{\mathrm{0}} {\int}^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \left(\mathrm{2}{x}\right)+\:{tan}^{\:−\mathrm{1}} \left(\frac{{x}}{\mathrm{2}}\:\right)}{\mathrm{1}+{x}^{\:\mathrm{2}} }{dx}=\frac{\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$

Question Number 159472 Answers: 1 Comments: 0

Question Number 159469 Answers: 1 Comments: 1

Question Number 159466 Answers: 2 Comments: 0

if tanA = (a/b) then prove that sinA = ± (a/( (√(a^ + b^ )))) please help..

$$\mathrm{if}\:{tanA}\:=\:\frac{{a}}{{b}} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:{sinA}\:=\:\pm\:\frac{{a}}{\:\sqrt{{a}^{ } \:+\:{b}^{ } }} \\ $$$$\mathrm{please}\:\mathrm{help}.. \\ $$

Question Number 159465 Answers: 1 Comments: 0

simplify ξ := Σ_(n=1) ^∞ ( (( 1)/(Σ_(k=1) ^n k^3 )) )=?

$$ \\ $$$$\:\:\:\:\:{simplify} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\xi\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\:\mathrm{1}}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\mathrm{3}} }\:\right)=? \\ $$$$ \\ $$

Question Number 159453 Answers: 1 Comments: 1

x and y are real value and x+iy=(5/(−3+4i)) find x and y.(i=(√(−1)))

$$\mathrm{x}\:{and}\:\mathrm{y}\:{are}\:{real}\:{value}\:{and} \\ $$$$\:\mathrm{x}+\mathrm{iy}=\frac{\mathrm{5}}{−\mathrm{3}+\mathrm{4}{i}}\:{find}\:\mathrm{x}\:{and}\:\mathrm{y}.\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 159461 Answers: 0 Comments: 4

≺PRIME-BIRTHDAYS≻ Do you know ′Prime1611′?... No,no it′s not an ID of the forum- member.It is a person who was born on November 16, 0001.On his birthday astrologers formed a string from his birthdate in the way: ′ddmmyyyy′: ′16110001′ The astro- logers observed that number containing in the string: 16110001 is a prime number.Thus they gave him name ′Prime1611′ They also suggested that Mr Prime1611 should celebrate his birthday only when the number ddmmyyyy be prime number.Recently He celebrated his birthday on 16-11-2021 as the number 16112021 is prime. How many birthdays has he celebrated upto his recent birthday when he had celebrated his first birthday on 16-11-0001? You may use calculator also. Question connected with Q#159421

$$\:\:\:\:\:\:\:\:\:\:\prec\mathbb{PRIME}-\mathcal{BIRTHDAYS}\succ \\ $$$$\mathrm{Do}\:\mathrm{you}\:\mathrm{know}\:'\mathrm{Prime1611}'?... \\ $$$$\mathrm{No},\mathrm{no}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{an}\:\mathrm{ID}\:\mathrm{of}\:\mathrm{the}\:\mathrm{forum}- \\ $$$$\mathrm{member}.\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{person}\:\mathrm{who}\:\mathrm{was}\:\mathrm{born} \\ $$$$\mathrm{on}\:\mathrm{November}\:\mathrm{16},\:\mathrm{0001}.\mathrm{On}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{astrologers}\:\mathrm{formed}\:\mathrm{a}\:\mathrm{string} \\ $$$$\mathrm{from}\:\mathrm{his}\:\mathrm{birthdate}\:\mathrm{in}\:\mathrm{the}\:\mathrm{way}: \\ $$$$'\mathrm{ddmmyyyy}':\:'\mathrm{16110001}'\:\mathrm{The}\:\mathrm{astro}- \\ $$$$\mathrm{logers}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{number}\:\mathrm{containing} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{string}:\:\mathrm{16110001}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime} \\ $$$$\mathrm{number}.\mathrm{Thus}\:\mathrm{they}\:\mathrm{gave}\:\mathrm{him}\:\mathrm{name} \\ $$$$'\mathrm{Prime1611}'\:\mathrm{They}\:\mathrm{also}\:\mathrm{suggested}\:\mathrm{that} \\ $$$$\mathrm{Mr}\:\mathrm{Prime1611}\:\mathrm{should}\:\mathrm{celebrate}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{ddmmyyyy}\:\mathrm{be}\:\mathrm{prime}\:\mathrm{number}.\mathrm{Recently} \\ $$$$\mathrm{He}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{2021}\:\mathrm{as}\:\mathrm{the}\:\mathrm{number}\:\mathrm{16112021} \\ $$$$\mathrm{is}\:\mathrm{prime}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{birthdays}\:\mathrm{has}\:\mathrm{he}\:\mathrm{celebrated} \\ $$$$\mathrm{upto}\:\mathrm{his}\:\mathrm{recent}\:\mathrm{birthday}\:\mathrm{when}\:\mathrm{he} \\ $$$$\mathrm{had}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{first}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{0001}? \\ $$$$\mathrm{You}\:\mathrm{may}\:\mathrm{use}\:\mathrm{calculator}\:\mathrm{also}. \\ $$$$\mathrm{Question}\:\mathrm{connected}\:\mathrm{with}\:\mathrm{Q}#\mathrm{159421} \\ $$

Question Number 159447 Answers: 1 Comments: 1

#calculate# Ω := ∫_0 ^( 1) ∫_0 ^( 1) (( x^( (t/2)) −x^( t) )/(1 − x)) dx dt = ? −−−m.n−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:#{calculate}# \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\:{x}^{\:\frac{{t}}{\mathrm{2}}} −{x}^{\:{t}} }{\mathrm{1}\:−\:{x}}\:{dx}\:{dt}\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:−−−{m}.{n}−−− \\ $$

Question Number 159428 Answers: 1 Comments: 0

(dy/dx)=cos(x+y)+sin(x+y)

$$\frac{{dy}}{{dx}}={cos}\left({x}+{y}\right)+{sin}\left({x}+{y}\right) \\ $$

Question Number 159425 Answers: 0 Comments: 0

Question Number 159421 Answers: 0 Comments: 4

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