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IntegrationQuestion and Answers: Page 93

Question Number 136651    Answers: 2   Comments: 0

Question Number 136644    Answers: 1   Comments: 0

𝛗=∫_0 ^( 1) ((ln(x^2 +1))/x^2 )dx f(a)=∫_0 ^( 1) ((log(ax^2 +1))/x^2 )dx f β€²(a)=∫_0 ^( 1) (x^2 /((ax^2 +1)x^2 ))dx=(1/a)∫_0 ^( 1) (dx/(x^2 +((1/( (√a))))^2 )) =((√a)/a)[tan^(βˆ’1) (x(√a) )]_0 ^1 =((√a)/a)tan^(βˆ’1) ((√a)) f(a)=^((√a) =u) ∫2tan^(βˆ’1) (u)du =2{u.tan^(βˆ’1) (u)βˆ’βˆ«(u/(1+u^2 ))du}+C =2(√a) tan^(βˆ’1) ((√a) )βˆ’ln(1+a)+C f(0)=0=0+Cβ‡’C=0 f(1)=𝛗=2((Ο€/4))βˆ’ln(2)=(Ο€/2)βˆ’ln(2)

Ο•=∫01ln(x2+1)x2dxf(a)=∫01log(ax2+1)x2dxfβ€²(a)=∫01x2(ax2+1)x2dx=1a∫01dxx2+(1a)2=aa[tanβˆ’1(xa)]01=aatanβˆ’1(a)f(a)=a=u∫2tanβˆ’1(u)du=2{u.tanβˆ’1(u)βˆ’βˆ«u1+u2du}+C=2atanβˆ’1(a)βˆ’ln(1+a)+Cf(0)=0=0+Cβ‡’C=0f(1)=Ο•=2(Ο€4)βˆ’ln(2)=Ο€2βˆ’ln(2)

Question Number 136626    Answers: 1   Comments: 0

..........nice calculus......... suppose that::: Ο•(p)=∫_0 ^( ∞) ((ln(1+x))/((p+x)^2 )) ...βœ“ find the value of:: ∫^( 1) _0 ((Ο•(p))/(1+p))dp=?...

..........nicecalculus.........supposethat:::Ο†(p)=∫0∞ln(1+x)(p+x)2...βœ“findthevalueof::∫01Ο†(p)1+pdp=?...

Question Number 136572    Answers: 3   Comments: 1

....advanced calculus.... 𝛗=∫_0 ^( (Ο€/2)) x.(tan(x))^(1/2) dx=??

....advancedcalculus....Ο•=∫0Ο€2x.(tan(x))12dx=??

Question Number 136497    Answers: 1   Comments: 2

......nice calculus..... prove:: ∫_0 ^( 1) (1/(1+ln^2 (x)))dx=∫_(0 ) ^( ∞) ((sin(x))/(1+x))dx

......nicecalculus.....prove::∫0111+ln2(x)dx=∫0∞sin(x)1+xdx

Question Number 136482    Answers: 0   Comments: 0

f(x)=∫_(βˆ’Ξ /4) ^(Π∫/4) e^(xtant) dt

f(x)=βˆ«βˆ’Ξ /4Π∫/4extantdt

Question Number 136481    Answers: 1   Comments: 0

∫_0 ^((50Ο€)/3) ∣sinx∣dx

∫050Ο€3∣sinx∣dx

Question Number 136476    Answers: 1   Comments: 0

(a) Let I(Ξ±)=∫_0 ^∞ e^(βˆ’(xβˆ’(Ξ±/x))^2 ) dx Show that it is legitimate to take the derivative of I(Ξ±) and also Iβ€²(Ξ±)= 0. Then show that I(Ξ±)=((βˆšΟ€)/2). (b) Use (a) to prove ∫_0 ^∞ e^(βˆ’(x^2 +Ξ±^2 x^(βˆ’2) )) dx=((βˆšΟ€)/2)e^(βˆ’2Ξ±) .

(a)LetI(Ξ±)=∫0∞eβˆ’(xβˆ’Ξ±x)2dxShowthatitislegitimatetotakethederivativeofI(Ξ±)andalsoIβ€²(Ξ±)=0.ThenshowthatI(Ξ±)=Ο€2.(b)Use(a)toprove∫0∞eβˆ’(x2+Ξ±2xβˆ’2)dx=Ο€2eβˆ’2Ξ±.

Question Number 136473    Answers: 0   Comments: 0

Question Number 136445    Answers: 2   Comments: 0

Question Number 136440    Answers: 2   Comments: 0

∫ (dx/(sin^6 x)) ?

∫dxsin6x?

Question Number 136425    Answers: 1   Comments: 3

If Ξ±>0 and Ξ²>0, prove ∫_0 ^∞ ((ln(Ξ±x))/(Ξ²^2 +x^2 ))dx=(Ο€/(2Ξ²))ln(Ξ±Ξ²)

IfΞ±>0andΞ²>0,prove∫0∞ln(Ξ±x)Ξ²2+x2dx=Ο€2Ξ²ln(Ξ±Ξ²)

Question Number 136406    Answers: 0   Comments: 1

calculate A_λ =∫_0 ^∞ ((cos^4 x)/((x^2 +λ^2 )^2 ))dx 2) find the value of ∫_0 ^∞ ((cos^4 x)/((x^2 +3)^2 ))dx

calculateAλ=∫0∞cos4x(x2+λ2)2dx2)findthevalueof∫0∞cos4x(x2+3)2dx

Question Number 136405    Answers: 0   Comments: 0

if f(x)=x^3 βˆ’3x+2 determine f^(βˆ’1) (x) and ∫ f^(βˆ’1) (nf(x))dx with n integr

iff(x)=x3βˆ’3x+2determinefβˆ’1(x)and∫fβˆ’1(nf(x))dxwithnintegr

Question Number 136403    Answers: 0   Comments: 0

find ∫ ((arctan(2x))/(x+3))dx

find∫arctan(2x)x+3dx

Question Number 136402    Answers: 0   Comments: 0

find U_n =∫_(1/n) ^n (1βˆ’(1/x^2 ))arctan(x+(1/x))dx and lim_(nβ†’βˆž) U_n

findUn=∫1nn(1βˆ’1x2)arctan(x+1x)dxandlimnβ†’βˆžUn

Question Number 136401    Answers: 1   Comments: 0

find ∫_0 ^∞ ((arctan(x^2 ))/(x^4 +1))dx

find∫0∞arctan(x2)x4+1dx

Question Number 136400    Answers: 0   Comments: 0

calculate ∫∫_([0,1]^2 ) e^(βˆ’(x^2 +y^2 )) arctan(x^2 +y^2 )dxdy

calculate∫∫[0,1]2eβˆ’(x2+y2)arctan(x2+y2)dxdy

Question Number 136399    Answers: 0   Comments: 0

calculate ∫ (dx/(x^n (√(x^2 βˆ’1))))

calculate∫dxxnx2βˆ’1

Question Number 136396    Answers: 0   Comments: 1

find ∫_0 ^1 (x^a /(1βˆ’x))dx

find∫01xa1βˆ’xdx

Question Number 136381    Answers: 1   Comments: 0

......sdvanced cslculus...... if x∈R^+ and:: 𝛗(x)=∫_0 ^( x) ((e^t βˆ’1)/t)ln((x/t))dt then prove that :: Ξ¨=∫_0 ^( ∞) e^(βˆ’x) 𝛗(x)dx=ΞΆ(2)

......sdvancedcslculus......ifx∈R+and::Ο•(x)=∫0xetβˆ’1tln(xt)dtthenprovethat::Ξ¨=∫0∞eβˆ’xΟ•(x)dx=ΞΆ(2)

Question Number 136365    Answers: 3   Comments: 0

∫ (dx/(sin x (√(cos x)))) =?

∫dxsinxcosx=?

Question Number 136343    Answers: 1   Comments: 0

lim_(xβ†’0) (x^(2021) /(xβˆ’ln(Ξ£_(k=0) ^(2020) (x^k /(k!))))) =^? 2021!

limxβ†’0x2021xβˆ’ln(βˆ‘2020k=0xkk!)=?2021!

Question Number 136333    Answers: 0   Comments: 0

calculate ∫_0 ^∞ (t^a /(1+t+t^2 ))dt study first the convergence (a real)

calculate∫0∞ta1+t+t2dtstudyfirsttheconvergence(areal)

Question Number 136325    Answers: 0   Comments: 1

please a generall Form for C(n) C(n)=(4/Ο€^2 )Ξ£_(k=1) ^n (βˆ’1)^(kβˆ’1) ΞΆ(2k)ΞΆ(2nβˆ’2k)

pleaseagenerallFormforC(n)C(n)=4Ο€2βˆ‘nk=1(βˆ’1)kβˆ’1ΞΆ(2k)ΞΆ(2nβˆ’2k)

Question Number 136279    Answers: 1   Comments: 0

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