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Question Number 108450 by bemath last updated on 17/Aug/20

  ((BeMath)/(⊂⊃))  (1)find ((1/2))!  (2)∫_0 ^(π/2) ((x sin x)/((1+cos x)^2 )) dx

BeMath⊂⊃(1)find(12)!(2)π/20xsinx(1+cosx)2dx

Commented by Smail last updated on 17/Aug/20

I_n =∫_0 ^∞ t^n e^(−t) dt=n!  n=(1/2)  I_(1/2) =∫_0 ^∞ (√t)e^(−t) dt=((1/2))!  x=(√t)⇒dx=(dt/(2(√t)))  I_(1/2) =2∫_0 ^∞ x^2 e^(−x^2 ) dx=((1/2))!  By parts  u=x⇒u′=1  v′=xe^(−x^2 ) ⇒v=((−1)/2)e^(−x^2 )   I_(1/2) =∫_0 ^∞ e^(−x^2 ) dx=((1/2))!  ∫_0 ^∞ e^(−x^2 ) dx=((√π)/2)  So  ((1/2))!=((√π)/2)

In=0tnetdt=n!n=12I1/2=0tetdt=(12)!x=tdx=dt2tI1/2=20x2ex2dx=(12)!Bypartsu=xu=1v=xex2v=12ex2I1/2=0ex2dx=(12)!0ex2dx=π2So(12)!=π2

Commented by bemath last updated on 17/Aug/20

thank you

thankyou

Answered by john santu last updated on 17/Aug/20

Answered by Dwaipayan Shikari last updated on 17/Aug/20

((1/2))!=(1/2)Γ((1/2))=((√π)/2)

(12)!=12Γ(12)=π2

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