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IntegrationQuestion and Answers: Page 107

Question Number 132693 Answers: 2 Comments: 0

I=∫ (dx/(x(x^2 +1)^3 ))

$$\mathrm{I}=\int\:\frac{{dx}}{{x}\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }\: \\ $$

Question Number 132610 Answers: 2 Comments: 0

Ω=∫ ((sin^2 (x))/(1+sin^2 (x))) dx

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

Question Number 132600 Answers: 0 Comments: 0

Question Number 132598 Answers: 0 Comments: 0

Find the voloume bounded by z=(√(x^2 +y^2 )) and the plane y+z=3

$$ \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{voloume}\:\mathrm{bounded}\:\mathrm{by}\:\mathrm{z}=\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} } \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{plane}\:\mathrm{y}+\mathrm{z}=\mathrm{3} \\ $$

Question Number 132534 Answers: 3 Comments: 1

∫_0 ^(π/2) (dx/(1+sin x)) →diverges or converges?

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{dx}}{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}\:\:\rightarrow\mathrm{diverges}\:\mathrm{or}\:\mathrm{converges}? \\ $$

Question Number 132519 Answers: 1 Comments: 0

.... nice calculus.... prove that :: Σ_(n=1) ^∞ (((−1)^n ln(n))/n)=γln(2)−(1/2)ln^2 (2)

$$\:\:\:\:\:....\:\:{nice}\:\:{calculus}.... \\ $$$$\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {ln}\left({n}\right)}{{n}}=\gamma{ln}\left(\mathrm{2}\right)−\frac{\mathrm{1}}{\mathrm{2}}{ln}^{\mathrm{2}} \left(\mathrm{2}\right) \\ $$$$ \\ $$

Question Number 132473 Answers: 1 Comments: 0

∫ ((x cosh x)/((sinh x)^2 )) dx

$$\int\:\frac{{x}\:\mathrm{cosh}\:{x}}{\left(\mathrm{sinh}\:{x}\right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 132469 Answers: 1 Comments: 0

Question Number 132463 Answers: 1 Comments: 0

Given ∫_a ^( b) ((x^2 −3x)/(∣x−3∣)) dx = ((11)/2) where { ((a<3<b)),((a+2b=8)) :} Find ∫_a ^b ∣x∣ dx.

$$\:\mathrm{Given}\:\int_{{a}} ^{\:{b}} \:\frac{{x}^{\mathrm{2}} −\mathrm{3}{x}}{\mid{x}−\mathrm{3}\mid}\:\mathrm{dx}\:=\:\frac{\mathrm{11}}{\mathrm{2}}\:\mathrm{where}\:\begin{cases}{{a}<\mathrm{3}<{b}}\\{{a}+\mathrm{2}{b}=\mathrm{8}}\end{cases} \\ $$$$\:\mathrm{Find}\:\int_{{a}} ^{{b}} \:\mid{x}\mid\:\mathrm{dx}.\: \\ $$

Question Number 132414 Answers: 1 Comments: 1

Question Number 132410 Answers: 4 Comments: 0

∫_0 ^( ∞) (dx/((x^4 −x^2 +1)^2 ))

$$\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{4}} −\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\: \\ $$

Question Number 132392 Answers: 0 Comments: 0

I = ∫e^(cos^(−1) x) dx = ?

$$\:\mathcal{I}\:=\:\int{e}^{\mathrm{cos}^{−\mathrm{1}} {x}} {dx}\:=\:? \\ $$

Question Number 132376 Answers: 0 Comments: 0

∫_0 ^(π/2) ((sin(nx))/(sinx))dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\left(\mathrm{nx}\right)}{\mathrm{sinx}}\mathrm{dx} \\ $$

Question Number 132333 Answers: 2 Comments: 0

very nice integral ∫ ((4x^3 +4x^2 +4x+3)/((x^2 +1)(x^2 +x+1)^2 )) dx?

$$\:\mathrm{very}\:\mathrm{nice}\:\mathrm{integral} \\ $$$$\int\:\frac{\mathrm{4x}^{\mathrm{3}} +\mathrm{4x}^{\mathrm{2}} +\mathrm{4x}+\mathrm{3}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx}? \\ $$$$ \\ $$

Question Number 132324 Answers: 1 Comments: 0

....advanced calculus... evaluation : 𝛗=∫_(0^( ) ) ^( ∞) ((ln(1+x))/(x(1+x^2 ))) dx solution: 𝛗=[∫_0 ^( 1) ((ln(1+x))/(x(1+x^2 )))dx=𝛗_1 ]+[∫_1 ^( ∞) ((ln(1+x))/(x(1+x^2 ))) dx=𝛗_2 ] 𝛗_2 =^(x=(1/t)) ∫_0 ^( 1) ((ln(1+(1/t)))/((1/t)(1+(1/t^2 ))))(dt/t^2 )=∫_0 ^( 1) ((tln(1+(1/t)))/(1+t^2 ))dt =∫_0 ^( 1) ((tln(1+t))/(1+t^2 ))dt−∫_0 ^( 1) ((tln(t))/(1+t^2 ))dt ∴ 𝛗=𝛗_1 +𝛗_2 =∫_0 ^( 1) ((1/x)+x)((ln(1+x))/(1+x^2 ))dx−Φ 𝛗=∫_0 ^( 1) ((ln(1+x))/x)dx−Φ=−li_2 (−1)−Φ Φ=∫_0 ^( 1) ((xln(x))/(1+x^2 ))dx=Σ_(n=0) ^∞ ∫_0 ^( 1) (−1)^n x^(2n+1) ln(x)dx =Σ_(n=0) ^∞ (−1)^n {[(x^(2n+2) /(2n+2)) ln(x)]_0 ^1 −(1/(2n+2))∫_0 ^( 1) x^(2n+1) dx} =Σ_(n=0) ^∞ (−1)^(n+1) (1/(4(n+1)^2 ))=−(1/4) Σ_(n=1) ^∞ (((−1)^(n−1) )/n^2 ) =((−1)/4) η(2)=((−π^2 )/(48)) .... ∴ 𝛗=−li_2 (−1)−Φ=(π^2 /(12))+(π^2 /(48)) 𝛗=((5π^2 )/(48))

$$\:\:\:\:\:\:\:\:\:\:\:\:....{advanced}\:\:\:{calculus}... \\ $$$$\:\:\:{evaluation}\:: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}\:^{\:\:} } ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:{dx} \\ $$$$\:\:\:\:{solution}: \\ $$$$\:\:\boldsymbol{\phi}=\left[\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}=\boldsymbol{\phi}_{\mathrm{1}} \right]+\left[\int_{\mathrm{1}} ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}\:{dx}=\boldsymbol{\phi}_{\mathrm{2}} \right] \\ $$$$\:\:\:\boldsymbol{\phi}_{\mathrm{2}} \overset{{x}=\frac{\mathrm{1}}{{t}}} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right)}{\frac{\mathrm{1}}{{t}}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)}\frac{{dt}}{{t}^{\mathrm{2}} }=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{tln}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{tln}\left(\mathrm{1}+{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}−\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{tln}\left({t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\therefore\:\:\boldsymbol{\phi}=\boldsymbol{\phi}_{\mathrm{1}} +\boldsymbol{\phi}_{\mathrm{2}} =\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}+{x}\right)\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}−\Phi \\ $$$$\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}\right)}{{x}}{dx}−\Phi=−{li}_{\mathrm{2}} \left(−\mathrm{1}\right)−\Phi \\ $$$$\:\Phi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{xln}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} {ln}\left({x}\right){dx} \\ $$$$\:\:=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \left\{\left[\frac{{x}^{\mathrm{2}{n}+\mathrm{2}} }{\mathrm{2}{n}+\mathrm{2}}\:{ln}\left({x}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} −\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}{n}+\mathrm{1}} {dx}\right\} \\ $$$$=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \frac{\mathrm{1}}{\mathrm{4}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }=−\frac{\mathrm{1}}{\mathrm{4}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\mathrm{2}} } \\ $$$$=\frac{−\mathrm{1}}{\mathrm{4}}\:\eta\left(\mathrm{2}\right)=\frac{−\pi^{\mathrm{2}} }{\mathrm{48}}\:\:\:.... \\ $$$$\:\:\:\:\therefore\:\boldsymbol{\phi}=−{li}_{\mathrm{2}} \left(−\mathrm{1}\right)−\Phi=\frac{\pi^{\mathrm{2}} }{\mathrm{12}}+\frac{\pi^{\mathrm{2}} }{\mathrm{48}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{48}} \\ $$$$\: \\ $$$$\:\:\: \\ $$

Question Number 132302 Answers: 1 Comments: 0

Simplify ((𝚪((p/2))𝚪((1/2)))/(𝚪((p/2)+(1/2))))

$$\boldsymbol{\mathrm{Simplify}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{\Gamma}\left(\frac{\boldsymbol{\mathrm{p}}}{\mathrm{2}}\right)\boldsymbol{\Gamma}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\boldsymbol{\Gamma}\left(\frac{\boldsymbol{\mathrm{p}}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$

Question Number 132301 Answers: 0 Comments: 0

..... nice.......calculus.... prove that :: ∫_0 ^( ∞) ((sin(2arctan((x/2))))/((x^2 +2^2 )sinh(πx)))dx=(7/8) −(π^2 /(12))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....\:{nice}.......{calculus}.... \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:\:{that}\::: \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{2}{arctan}\left(\frac{{x}}{\mathrm{2}}\right)\right)}{\left({x}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \right){sinh}\left(\pi{x}\right)}{dx}=\frac{\mathrm{7}}{\mathrm{8}}\:−\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$$$ \\ $$

Question Number 132235 Answers: 1 Comments: 0

I= ∫ ((3x+5)/((x^2 +2x+3)^2 )) dx ?

$$\:\mathrm{I}=\:\int\:\frac{\mathrm{3x}+\mathrm{5}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{3}\right)^{\mathrm{2}} }\:\mathrm{dx}\:? \\ $$

Question Number 132287 Answers: 2 Comments: 0

Question Number 132192 Answers: 3 Comments: 0

∫ (dx/(csc x + sec x))

$$\:\int\:\frac{\mathrm{dx}}{\mathrm{csc}\:\mathrm{x}\:+\:\mathrm{sec}\:\mathrm{x}} \\ $$

Question Number 132191 Answers: 2 Comments: 0

∫_0 ^1 ((log_e (x+1))/(x(x^2 +1)))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{log}_{{e}} \left({x}+\mathrm{1}\right)}{{x}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{dx} \\ $$

Question Number 132121 Answers: 2 Comments: 2

∫_2 ^8 ((√x)/( (√(10−x)) +(√x))) dx

$$\int_{\mathrm{2}} ^{\mathrm{8}} \frac{\sqrt{{x}}}{\:\sqrt{\mathrm{10}−{x}}\:+\sqrt{{x}}}\:\mathrm{dx} \\ $$

Question Number 132110 Answers: 1 Comments: 0

∫(1/(ln x))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{ln}\:{x}}{dx}=? \\ $$

Question Number 132091 Answers: 2 Comments: 0

evaluate ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ((da.db.dc.dd.df)/(1−abcdf))

$${evaluate}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{da}.{db}.{dc}.{dd}.{df}}{\mathrm{1}−{abcdf}} \\ $$

Question Number 132090 Answers: 3 Comments: 0

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

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

Question Number 132089 Answers: 1 Comments: 0

∫_0 ^∞ (1/((x^4 −x^2 +1)^3 ))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left({x}^{\mathrm{4}} −{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

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