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Question Number 137311    Answers: 1   Comments: 0

Question Number 137307    Answers: 1   Comments: 0

.....advanced calculus..... prove that:: Σ_(n=0) ^∞ ((((−1)^n )/(4n^2 +1)))=(1/4)(2+πcsch((π/2))) ............................

$$\:\:\:\:\:\:\:\:.....{advanced}\:\:\:\:{calculus}..... \\ $$$$\:\:\:\:{prove}\:\:{that}:: \\ $$$$\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}{n}^{\mathrm{2}} +\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{2}+\pi{csch}\left(\frac{\pi}{\mathrm{2}}\right)\right) \\ $$$$\:\:\:\:\:\:............................ \\ $$

Question Number 137303    Answers: 0   Comments: 2

from the figure above the square S′s diameter length is increasing by 25 m/s to the north−east initially at length 30(√(2 ))m and circle C′s radius is decreasing by 2 m/s initially at length 100 m knowing that the blue line′s length = 40m at what time the horizontal distance between point p and point q will equal 0? and what will be the vertical distance at that time?

$$\mathrm{from}\:\mathrm{the}\:\mathrm{figure}\:\mathrm{above} \\ $$$${the}\:{square}\:{S}'\mathrm{s}\:\mathrm{diameter}\:\mathrm{length}\:\mathrm{is}\:\mathrm{increasing} \\ $$$$\mathrm{by}\:\mathrm{25}\:\mathrm{m}/\mathrm{s}\:\mathrm{to}\:\mathrm{the}\:\mathrm{north}−\mathrm{east}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{length}\:\mathrm{30}\sqrt{\mathrm{2}\:}\mathrm{m}\:\mathrm{and}\:\mathrm{circle}\: \\ $$$${C}'\mathrm{s}\:\mathrm{radius}\:\mathrm{is}\:\mathrm{decreasing}\:\mathrm{by}\:\mathrm{2}\:\mathrm{m}/\mathrm{s}\:\mathrm{initially}\:\mathrm{at}\:\mathrm{length}\:\mathrm{100}\:\mathrm{m} \\ $$$$\mathrm{knowing}\:\mathrm{that}\:\mathrm{the}\:\mathrm{blue}\:\mathrm{line}'\mathrm{s}\:\mathrm{length}\:=\:\mathrm{40m} \\ $$$$\mathrm{at}\:\mathrm{what}\:\mathrm{time}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{point}\:{p} \\ $$$$\mathrm{and}\:\mathrm{point}\:{q}\:\mathrm{will}\:\mathrm{equal}\:\mathrm{0}? \\ $$$$\mathrm{and}\:\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{distance}\:\mathrm{at}\:\mathrm{that}\:\mathrm{time}? \\ $$$$ \\ $$

Question Number 137302    Answers: 0   Comments: 0

∫(√(x+(√(x+(√(x+(√x))))))) dx

$$\int\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}}\:{dx} \\ $$

Question Number 137300    Answers: 0   Comments: 0

Question Number 137290    Answers: 2   Comments: 0

Question Number 137285    Answers: 3   Comments: 0

∫ ((5+2cos 2x)/(4+2sin 2x)) dx

$$\int\:\frac{\mathrm{5}+\mathrm{2cos}\:\mathrm{2x}}{\mathrm{4}+\mathrm{2sin}\:\mathrm{2x}}\:\mathrm{dx}\: \\ $$

Question Number 137282    Answers: 2   Comments: 0

Question Number 137279    Answers: 1   Comments: 0

∫(dx/(sin (a+x)cos (b+x)))=?

$$\int\frac{{dx}}{\mathrm{sin}\:\left({a}+{x}\right)\mathrm{cos}\:\left({b}+{x}\right)}=? \\ $$

Question Number 137280    Answers: 2   Comments: 0

Question Number 137277    Answers: 3   Comments: 0

𝛗=∫_0 ^( (π/2)) sin^2 (x).ln(sin(x))dx=?

$$ \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}} \left({x}\right).{ln}\left({sin}\left({x}\right)\right){dx}=? \\ $$$$ \\ $$

Question Number 137275    Answers: 0   Comments: 0

......complex analysis..... if , f(α,n,x)=(d^( n) /dx^n )(α^x ) , x∈C α∈C−{0} , n∈C−Z^− ∪{0} and g(n,x)=∫_0 ^( 1) f(α,n,x)dα then find the value of ... Ω=((Im(g(i,0)))/(Re(Γ(i)))) solution: g(n,x)=∫_0 ^( 1) (d^( n) /dx^(n ) )(α^x )dα =∫_0 ^( 1) (d^( n) /dx^n )(e^(xln(α)) )dα .....⟨∗⟩ (d^( n) /dx^n )(e^(xln(α)) )=(ln(α))^n α^x ⟨∗⟩→ ...g(n,x)=∫_0 ^( 1) (ln(α))^n α^x dα =_(α=e^(−y) ) ^(ln(α)=−y) ∫_0 ^( ∞) (−1)^n y^( n) e^(−yx) e^(−y) dy =(−1)^n ∫_0 ^( 1) y^n .e^(−y(1+x)) dy =^(y(1+x)=t) (−1)^n ∫_0 ^( 1) ((t^n e^(−t) )/((1+x)^(n+1) ))dt =e^(inπ) .(1/((1+x)^(n+1) )) .Γ(n+1) g(i,0)=e^(−π) .i.Γ(i) ......⟨∗∗⟩ Γ(i)∈C ⇒ Γ(i)=Re(Γ(i))+Im(Γ(i)) ⟨∗∗⟩→ ... g(i,0)=e^(−π) .i.[Re(Γ(i))+Im(Γ(i))] ∴ Ω=((e^(−π) Re(Γ(i)))/(Re(Γ(i)))) =e^(−π) ...✓✓

$$\:\:\:\:\:\:\:\:\:\:\:\:\:......{complex}\:\:{analysis}..... \\ $$$$\:\:\:\:{if}\:,\:\:{f}\left(\alpha,{n},{x}\right)=\frac{{d}^{\:{n}} }{{dx}^{{n}} }\left(\alpha^{{x}} \right)\:\:,\:{x}\in\mathbb{C} \\ $$$$\:\:\:\:\:\alpha\in\mathbb{C}−\left\{\mathrm{0}\right\}\:,\:{n}\in\mathbb{C}−\mathbb{Z}^{−} \cup\left\{\mathrm{0}\right\} \\ $$$$\:\:\:\:\:{and}\:\:{g}\left({n},{x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {f}\left(\alpha,{n},{x}\right){d}\alpha \\ $$$$\:\:\:\:\:{then}\:\:{find}\:\:{the}\:{value}\:{of}\:... \\ $$$$\:\:\:\:\:\:\Omega=\frac{{Im}\left({g}\left({i},\mathrm{0}\right)\right)}{{Re}\left(\Gamma\left({i}\right)\right)} \\ $$$$\:\:\:\:\:\:{solution}: \\ $$$$\:\:\:\:\:\:{g}\left({n},{x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{d}^{\:{n}} }{{dx}^{{n}\:} }\left(\alpha^{{x}} \right){d}\alpha \\ $$$$\:\:\:\:\:\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{d}^{\:{n}} }{{dx}^{{n}} }\left({e}^{{xln}\left(\alpha\right)} \right){d}\alpha\:.....\langle\ast\rangle \\ $$$$\:\:\:\:\:\:\:\frac{{d}^{\:{n}} }{{dx}^{{n}} }\left({e}^{{xln}\left(\alpha\right)} \right)=\left({ln}\left(\alpha\right)\right)^{{n}} \alpha^{{x}} \\ $$$$\:\:\:\langle\ast\rangle\rightarrow\:...{g}\left({n},{x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left({ln}\left(\alpha\right)\right)^{{n}} \alpha^{{x}} {d}\alpha \\ $$$$\:\:\underset{\alpha={e}^{−{y}} } {\overset{{ln}\left(\alpha\right)=−{y}} {=}}\int_{\mathrm{0}} ^{\:\infty} \left(−\mathrm{1}\right)^{{n}} {y}^{\:{n}} {e}^{−{yx}} {e}^{−{y}} {dy} \\ $$$$\:\:\:\:\:=\left(−\mathrm{1}\right)^{{n}} \int_{\mathrm{0}} ^{\:\mathrm{1}} {y}^{{n}} .{e}^{−{y}\left(\mathrm{1}+{x}\right)} {dy} \\ $$$$\:\:\:\:\overset{{y}\left(\mathrm{1}+{x}\right)={t}} {=}\left(−\mathrm{1}\right)^{{n}} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{t}^{{n}} \:{e}^{−{t}} }{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} }{dt} \\ $$$$\:\:\:\:\:\:\:={e}^{{in}\pi} .\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} }\:.\Gamma\left({n}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:{g}\left({i},\mathrm{0}\right)={e}^{−\pi} .{i}.\Gamma\left({i}\right)\:......\langle\ast\ast\rangle \\ $$$$\:\:\:\Gamma\left({i}\right)\in\mathbb{C}\:\Rightarrow\:\:\Gamma\left({i}\right)={Re}\left(\Gamma\left({i}\right)\right)+{Im}\left(\Gamma\left({i}\right)\right) \\ $$$$\:\:\:\langle\ast\ast\rangle\rightarrow\:...\:{g}\left({i},\mathrm{0}\right)={e}^{−\pi} .{i}.\left[{Re}\left(\Gamma\left({i}\right)\right)+{Im}\left(\Gamma\left({i}\right)\right)\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\Omega=\frac{{e}^{−\pi} {Re}\left(\Gamma\left({i}\right)\right)}{{Re}\left(\Gamma\left({i}\right)\right)}\:={e}^{−\pi} ...\checkmark\checkmark \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 137269    Answers: 1   Comments: 1

Question Number 137255    Answers: 2   Comments: 0

Question Number 137252    Answers: 1   Comments: 0

Decompose the function P(x) = ((x^4 +2x^3 +6x^2 +20x+6)/(x^3 +x^2 +x)) in partial fractions.

$$\mathrm{Decompose}\:\mathrm{the}\:\mathrm{function}\:\mathrm{P}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}^{\mathrm{4}} +\mathrm{2x}^{\mathrm{3}} +\mathrm{6x}^{\mathrm{2}} +\mathrm{20x}+\mathrm{6}}{\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\: \\ $$$$\mathrm{in}\:\mathrm{partial}\:\mathrm{fractions}. \\ $$

Question Number 137251    Answers: 2   Comments: 0

......Advanced ... calculus...... 𝛗=∫_0 ^( 1) x^2 ln(x)ln(1−x)dx=???

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:......\mathscr{A}{dvanced}\:\:...\:\:{calculus}...... \\ $$$$\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}} {ln}\left({x}\right){ln}\left(\mathrm{1}−{x}\right){dx}=??? \\ $$$$ \\ $$

Question Number 137249    Answers: 5   Comments: 0

∫ (((3sin x+2))/((2sin x+3)^2 )) dx =?

$$\int\:\frac{\left(\mathrm{3sin}\:\mathrm{x}+\mathrm{2}\right)}{\left(\mathrm{2sin}\:\mathrm{x}+\mathrm{3}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 137226    Answers: 1   Comments: 0

Question Number 137225    Answers: 0   Comments: 0

Q137026

$${Q}\mathrm{137026} \\ $$

Question Number 137217    Answers: 0   Comments: 0

Q137014

$${Q}\mathrm{137014} \\ $$

Question Number 137210    Answers: 1   Comments: 0

Question Number 137209    Answers: 1   Comments: 0

Question Number 137208    Answers: 0   Comments: 0

If x,y > 0 then prove that 3(√((x^2 +y^2 )/2)) + (√(xy)) ≥ 2(x+y)

$$\mathrm{If}\:\mathrm{x},\mathrm{y}\:>\:\mathrm{0}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\: \\ $$$$\:\mathrm{3}\sqrt{\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\:+\:\sqrt{\mathrm{xy}}\:\geqslant\:\mathrm{2}\left(\mathrm{x}+\mathrm{y}\right)\: \\ $$

Question Number 137206    Answers: 2   Comments: 0

Question Number 137203    Answers: 2   Comments: 0

∫ ((ln(1+x))/x)=?

$$\int\:\frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}}=? \\ $$

Question Number 137197    Answers: 1   Comments: 0

α , β ε (0 , (π/2)) tan^2 α = 1+2tan^2 β ⇒(√2)cosα−cosβ=?

$$\alpha\:,\:\beta\:\epsilon\:\left(\mathrm{0}\:,\:\frac{\pi}{\mathrm{2}}\right) \\ $$$${tan}^{\mathrm{2}} \alpha\:=\:\mathrm{1}+\mathrm{2}{tan}^{\mathrm{2}} \beta\:\:\Rightarrow\sqrt{\mathrm{2}}{cos}\alpha−{cos}\beta=? \\ $$

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