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IntegrationQuestion and Answers: Page 77

Question Number 140634 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) ((cos(2sinx))/((x^2 −x+1)^2 ))dx

$${find}\:\int_{−\infty} ^{+\infty} \:\frac{{cos}\left(\mathrm{2}{sinx}\right)}{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 140635 Answers: 1 Comments: 0

find ∫_(−∞) ^(+∞) ((sin(2cosx))/((x^2 −x+1)^2 ))dx

$${find}\:\:\int_{−\infty} ^{+\infty} \:\frac{{sin}\left(\mathrm{2}{cosx}\right)}{\left({x}^{\mathrm{2}} \:−{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 140615 Answers: 1 Comments: 0

Find the area common to the curve y^2 = 12x and x^2 +y^2 = 24x .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{common}\: \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{12x}\:\mathrm{and} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\:\mathrm{24x}\:. \\ $$

Question Number 140614 Answers: 2 Comments: 0

∫ _0^(π/2) ln (sin x) sec^2 x dx =?

$$\int\:_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{ln}\:\left(\mathrm{sin}\:\mathrm{x}\right)\:\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:=?\: \\ $$$$ \\ $$

Question Number 140588 Answers: 1 Comments: 0

.......Advanced ....★★★....Calculus....... evaluation the value of : 𝛗 :=∫_0 ^( (π/2)) sin^2 (x).ln(sin(x))dx solution:: ξ (a):=∫_0 ^( (π/2)) sin^(2+a) (x)dx =(1/2)β (((3+a)/2) ,(1/2)) :=(1/2)(((Γ(((3+a)/2))Γ((1/2)))/(Γ(2+(a/2))))).......✓ 𝛗:= ξ ′ (0) ..............✓ :=(1/2) (√π) (( Γ′(((3+a)/2)).Γ(2+(a/2))−Γ(((3+a)/2)).Γ′(2+(a/2)))/(Γ^2 (2+(a/2)))) ∣_(a=0) :=(1/2)(√π) ((Γ′((3/2))−Γ((3/2)).Γ′(2))/(( Γ^2 (2):=1 ))) :=(1/2)(√π) ((ψ((3/2))Γ((3/2))−Γ((3/2)).ψ(2))/1) := ((√π)/4){ (2−γ−2ln(2)−(1−γ)} :=((√π)/4)(1−ln(4))=(√π) ln(((e/4))^(1/4) )

Question Number 140554 Answers: 0 Comments: 5

What′s the relationship between Dirichlet β(s) function with ζ(s) function ? That is Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^s )) with Σ_(n=1) ^∞ (1/n^s ).

$${What}'{s}\:{the}\:{relationship}\:{between}\:{Dirichlet}\:\beta\left({s}\right)\:{function}\:{with} \\ $$$$\zeta\left({s}\right)\:{function}\:?\:{That}\:{is}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{{s}} }\:\:{with}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }. \\ $$

Question Number 140500 Answers: 0 Comments: 0

tan^2 1°+tan^2 2°+tan^2 3°+...+tan^2 89°=((15931)/3) ???

$$\mathrm{tan}\:^{\mathrm{2}} \mathrm{1}°+\mathrm{tan}\:^{\mathrm{2}} \mathrm{2}°+\mathrm{tan}\:^{\mathrm{2}} \mathrm{3}°+...+\mathrm{tan}\:^{\mathrm{2}} \mathrm{89}°=\frac{\mathrm{15931}}{\mathrm{3}}\:\:\:\:\:??? \\ $$

Question Number 140490 Answers: 1 Comments: 1

Question Number 140447 Answers: 1 Comments: 0

Question Number 140405 Answers: 1 Comments: 0

find the value of :: Θ :=Σ_(n=1 ) ^∞ (1/(4n.(4n+1).(4n+2).(4n+3)))=?

$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:{find}\:\:{the}\:\:{value}\:{of}\::: \\ $$$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\Theta\::=\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{4}{n}.\left(\mathrm{4}{n}+\mathrm{1}\right).\left(\mathrm{4}{n}+\mathrm{2}\right).\left(\mathrm{4}{n}+\mathrm{3}\right)}=? \\ $$$$\:\:\:\:\: \\ $$

Question Number 140401 Answers: 2 Comments: 0

evaluate :: Φ:=∫_0 ^( ∞) xe^(−(x^2 /4)) ln(x)dx = m.( π γ) find ” m ” ......

$$\:\:\: \\ $$$$\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\Phi:=\int_{\mathrm{0}} ^{\:\infty} {xe}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}} {ln}\left({x}\right){dx}\:=\:{m}.\left(\:\pi\:\gamma\right) \\ $$$$\:\:\:\:\:\:{find}\:\:\:''\:\:{m}\:\:''\:...... \\ $$$$ \\ $$

Question Number 140399 Answers: 1 Comments: 0

𝛏 :=∫_0 ^( ∞) ((e^(−x^2 ) −e^(−x) )/x) dx = k.γ find ” k ” ... γ := Euler constant....

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\xi}\::=\int_{\mathrm{0}} ^{\:\infty} \:\frac{{e}^{−{x}^{\mathrm{2}} } −{e}^{−{x}} }{{x}}\:{dx}\:=\:{k}.\gamma\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{find}\:\:''\:{k}\:\:''\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\gamma\::=\:\mathscr{E}{uler}\:{constant}.... \\ $$

Question Number 140388 Answers: 3 Comments: 0

∫_0 ^∞ ((ln x)/((x^2 +a^2 )^5 )) dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{ln}\:\mathrm{x}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{a}^{\mathrm{2}} \right)^{\mathrm{5}} }\:\mathrm{dx}\: \\ $$

Question Number 140353 Answers: 0 Comments: 1

Question Number 140350 Answers: 0 Comments: 0

Question Number 140334 Answers: 1 Comments: 0

calculate :: 𝛏 := ∫_(−∞) ^( ∞) ln(2−2cos(x^2 ))dx=?

$$ \\ $$$$\:\:\:\:\:{calculate}\::: \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\xi}\::=\:\int_{−\infty} ^{\:\infty} {ln}\left(\mathrm{2}−\mathrm{2}{cos}\left({x}^{\mathrm{2}} \right)\right){dx}=? \\ $$$$ \\ $$

Question Number 140330 Answers: 2 Comments: 0

Find the Integration Value: 1 ∫(((√x)d(x))/(1+^3 (√x)))=? 2 ∫(dx/(x^(1/2) −x^(1/4) ))=?

$${Find}\:{the}\:{Integration}\:{Value}: \\ $$$$\mathrm{1} \:\int\frac{\sqrt{{x}}{d}\left({x}\right)}{\mathrm{1}+^{\mathrm{3}} \sqrt{{x}}}=? \\ $$$$\mathrm{2} \int\frac{{dx}}{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} −{x}^{\frac{\mathrm{1}}{\mathrm{4}}} }=? \\ $$

Question Number 140310 Answers: 2 Comments: 0

prove that ∫_0 ^∞ ((ln x)/(x^2 +1)) dx = 0

$$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{dx}\:=\:\mathrm{0} \\ $$

Question Number 140282 Answers: 0 Comments: 0

Σ_(k=0) ^(p−1) ((p),(k) )sin [2(p−k)x]=? ((p),(0) )sin (2px)+ ((p),(1) )sin [(2p−2)x]+ ((p),(2) )sin [(2p−4)x]+...+ ((( p)),((p−1)) )sin (2x)=2^p ∙cos^p (x)∙sin (px) ??? or ∫_0 ^∞ ((cos^p (x)∙sin (px))/x)dx=(π/2)(1−2^(−p) ) why ???

$$\underset{{k}=\mathrm{0}} {\overset{{p}−\mathrm{1}} {\sum}}\begin{pmatrix}{{p}}\\{{k}}\end{pmatrix}\mathrm{sin}\:\left[\mathrm{2}\left({p}−{k}\right){x}\right]=? \\ $$$$\begin{pmatrix}{{p}}\\{\mathrm{0}}\end{pmatrix}\mathrm{sin}\:\left(\mathrm{2}{px}\right)+\begin{pmatrix}{{p}}\\{\mathrm{1}}\end{pmatrix}\mathrm{sin}\:\left[\left(\mathrm{2}{p}−\mathrm{2}\right){x}\right]+\begin{pmatrix}{{p}}\\{\mathrm{2}}\end{pmatrix}\mathrm{sin}\:\left[\left(\mathrm{2}{p}−\mathrm{4}\right){x}\right]+...+\begin{pmatrix}{\:\:\:{p}}\\{{p}−\mathrm{1}}\end{pmatrix}\mathrm{sin}\:\left(\mathrm{2}{x}\right)=\mathrm{2}^{{p}} \centerdot\mathrm{cos}\:^{{p}} \left({x}\right)\centerdot\mathrm{sin}\:\left({px}\right)\:\:\:\:\:??? \\ $$$${or}\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\:^{{p}} \left({x}\right)\centerdot\mathrm{sin}\:\left({px}\right)}{{x}}{dx}=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\mathrm{2}^{−{p}} \right)\:\:\:\:\:{why}\:??? \\ $$

Question Number 140278 Answers: 0 Comments: 0

∫(√(x (/)))

$$\int\sqrt{{x}\:\frac{}{}} \\ $$

Question Number 140221 Answers: 1 Comments: 0

∫_0 ^∞ x^2 [ln(1+e^x )−x]dx=((7π^4 )/(360))

$$\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} \left[{ln}\left(\mathrm{1}+{e}^{{x}} \right)−{x}\right]{dx}=\frac{\mathrm{7}\pi^{\mathrm{4}} }{\mathrm{360}} \\ $$

Question Number 140202 Answers: 2 Comments: 0

∫_0 ^∞ (((lnx)/(x−1)))^2 dx=(2/3)π^2

$$\int_{\mathrm{0}} ^{\infty} \left(\frac{{lnx}}{{x}−\mathrm{1}}\right)^{\mathrm{2}} {dx}=\frac{\mathrm{2}}{\mathrm{3}}\pi^{\mathrm{2}} \\ $$

Question Number 140200 Answers: 1 Comments: 1

∫_0 ^∞ (((lnx)/(x−1)))^3 dx=π^2