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intregation-collected-problem-




Question Number 124076 by TANMAY PANACEA last updated on 30/Nov/20
intregation...collected problem
intregationcollectedproblem
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
calculus collected roblem
calculuscollectedroblem
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by TANMAY PANACEA last updated on 30/Nov/20
Commented by Dwaipayan Shikari last updated on 30/Nov/20
∫_0 ^a (dx/((a^n −x^n )^(1/n) ))=(1/a)∫_0 ^a (1/((1−(x^n /a^n ))^(1/n) ))dx   u=(x/a) ⇒  =∫_0 ^1 (1/((1−u^n )^(1/n) ))du =(1/n)∫_0 ^1 u^(1−n) (1−t)^(−(1/n)) dt               u^n =t⇒nu^(n−1) =(dt/du)  =(1/n)∫_0 ^1 t^((1/n)−1) (1−t)^(−(1/n)) =(1/n).((Γ((1/n))Γ(1−(1/n)))/(Γ(1)))=(π/(nsin((π/n))))
0adx(anxn)1n=1a0a1(1xnan)1ndxu=xa=011(1un)1ndu=1n01u1n(1t)1ndtun=tnun1=dtdu=1n01t1n1(1t)1n=1n.Γ(1n)Γ(11n)Γ(1)=πnsin(πn)
Commented by TANMAY PANACEA last updated on 30/Nov/20
thank you but question no 22 pls
thankyoubutquestionno22pls
Commented by mnjuly1970 last updated on 30/Nov/20
    note:: Γ(x)Γ(1−x)=(π/(sin(πx)))   note::  ∫_0 ^( π) ln(sin(x))dx=−πln(2)  Ω=∫_0 ^( 1) ln(Γ(x))dx  Ω=∫_0 ^( 1) ln(Γ(1−x))dx   2Ω=∫_0 ^( 1) ln[(Γ(x)Γ(1−x))]dx    2Ω =^(euler reflection formula) ∫_0 ^( 1) ln((π/(sin(πx))))dx  =ln(π)−∫_0 ^( 1) ln(sin(πx))dx  =^(πx=t) ln(π)−(1/π)∫_0 ^( π) ln(sin(x))dx  =ln(π)−(1/π)(−πln(2))  =ln(π)+ln(2)=ln(2π)  ∴  Ω=(1/2)ln(2π)=ln((√(2π)) )✓
note::Γ(x)Γ(1x)=πsin(πx)note::0πln(sin(x))dx=πln(2)Ω=01ln(Γ(x))dxΩ=01ln(Γ(1x))dx2Ω=01ln[(Γ(x)Γ(1x))]dx2Ω=eulerreflectionformula01ln(πsin(πx))dx=ln(π)01ln(sin(πx))dx=πx=tln(π)1π0πln(sin(x))dx=ln(π)1π(πln(2))=ln(π)+ln(2)=ln(2π)Ω=12ln(2π)=ln(2π)
Commented by TANMAY PANACEA last updated on 30/Nov/20
excellent
excellent
Commented by mnjuly1970 last updated on 30/Nov/20
you are welcom sir tanmay..
youarewelcomsirtanmay..
Commented by Dwaipayan Shikari last updated on 30/Nov/20
Γ(s)Γ(1−s)=(π/(sinπs))  log(Γ(s)+log(Γ(1−s))=log(π)−log(sinπs)  ∫_0 ^1 log(Γ(s))=∫_0 ^1 log(Γ(1−s))ds  ∫_0 ^1 2log(Γ(s))=(1/2)∫_0 ^1 log(π)ds−(1/2)∫_0 ^1 log(sinπs)ds  ∫_0 ^1 log(Γ(s))=(1/2)log(π)−(1/(2π))∫_0 ^π log(sinu)du     u=2t⇒1=2(dt/du)  =(1/2)log(π)−(2/(2π))∫_0 ^(π/2) log(2)+log(sint)+log(cost)  =(1/2)log(π)−(1/2)log(2)+(1/2)log(4)  =(1/2)log(2π)
Γ(s)Γ(1s)=πsinπslog(Γ(s)+log(Γ(1s))=log(π)log(sinπs)01log(Γ(s))=01log(Γ(1s))ds012log(Γ(s))=1201log(π)ds1201log(sinπs)ds01log(Γ(s))=12log(π)12π0πlog(sinu)duu=2t1=2dtdu=12log(π)22π0π2log(2)+log(sint)+log(cost)=12log(π)12log(2)+12log(4)=12log(2π)
Commented by Dwaipayan Shikari last updated on 30/Nov/20
∫_0 ^(π/2) (1/( (√(acos^4 x+bsin^4 x))))dx  =∫_0 ^(π/2) ((sec^2 x)/( (√(a+btan^4 x))))dx=∫_0 ^∞ (dt/( (√(a+bt^4 ))))                  a+bt^4 =u^2 ⇒2bt^3 =u(du/dt)  =(1/2)∫_(√a) ^∞ ((udu)/(ubt^3 ))=(1/(2b))∫_(√a) ^∞ (du/((((u^2 −a)/b))^(3/4) ))  =(1/(2 (b)^(1/4) ))∫_0 ^∞ (du/((u^2 −a)^(3/4) ))                      u^2 −a=p⇒2u=(dp/du)  =(1/(4(b)^(1/4) ))∫_0 ^∞ (dp/(p^(3/4) (a+p)^(1/2) ))=(1/(4((ab))^(1/4) ))∫_0 ^∞ (1/(Φ^(3/4) (1+Φ)^(1/2) ))dΦ      Φ=(p/a)  =(1/(4((ab))^(1/4) ))∫_0 ^1 Λ^(−(3/4)) (1−Λ)^(−(3/4)) dΛ                (Φ/(1+Φ))=Λ  =(1/(4((ab))^(1/4) )).Γ^2 ((1/4)).(1/(Γ((1/2))))=((Γ^2 ((1/4)))/(4((π^2 ab))^(1/4) ))
0π21acos4x+bsin4xdx=0π2sec2xa+btan4xdx=0dta+bt4a+bt4=u22bt3=ududt=12auduubt3=12badu(u2ab)34=12b40du(u2a)34u2a=p2u=dpdu=14b40dpp34(a+p)12=14ab401Φ34(1+Φ)12dΦΦ=pa=14ab401Λ34(1Λ)34dΛΦ1+Φ=Λ=14ab4.Γ2(14).1Γ(12)=Γ2(14)4π2ab4

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