Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 99707 by student work last updated on 22/Jun/20

∫_(−∞) ^∞ e^(−x^2 ) dx=?

ex2dx=?

Commented by student work last updated on 22/Jun/20

help me

helpme

Answered by smridha last updated on 22/Jun/20

2∫_0 ^∞ e^(−x^2 ) x^(2.(1/2)−1) dx  =𝚪((1/2))=(√𝛑)

20ex2x2.121dx=Γ(12)=π

Answered by smridha last updated on 22/Jun/20

∫_(−∞) ^∞ e^(−x^2 ) dx=?  let′s do it another way  I=∫_(−∞) ^(+∞) e^(−x^2 ) dx   similarly I=∫_(−∞) ^(+∞) e^(−y^2 ) dy  so now I^2 =∫_(−∞) ^(+∞) ∫_(−∞) ^(+∞) e^(−(x^2 +y^2 )) dxdy  now by transforming to this  integral cartesian to plane−polar  co−ordinate we get  I^2 =∫_0 ^(2𝛑) d𝛉 ∫_0 ^∞ e^(−r^2 ) rdr=2𝛑.(−(1/2))[e^(−r^2 ) ]_0 ^∞                   =𝛑  so I=(√𝛑) (consider positive root).    this is one of the valuable integral.

ex2dx=?letsdoitanotherwayI=+ex2dxsimilarlyI=+ey2dysonowI2=++e(x2+y2)dxdynowbytransformingtothisintegralcartesiantoplanepolarcoordinatewegetI2=02πdθ0er2rdr=2π.(12)[er2]0=πsoI=π(considerpositiveroot).thisisoneofthevaluableintegral.

Answered by mathmax by abdo last updated on 23/Jun/20

  A =∫_(−∞) ^(+∞)  e^(−x^2 ) dx ⇒A =2 ∫_0 ^∞  e^(−x^2 ) dx  let  A_n =∫∫_([(1/n),n[^2 )   e^(−x^2 −y^2 ) dxdy  we have A_n =∫_((1 )/n) ^(n )   e^(−x^2 ) dx.∫_(1/n) ^n  e^(−y^2 ) dy ⇒lim_(n→+∞) A_n =(∫_0 ^∞  e^(−x^2 ) dx)^2   let considere the diffeomorphisme  { ((x = r cosθ)),((y =rsinθ)) :}  (1/n)≤x <n and (1/n)≤y<n ⇒(2/n^2 )≤x^2  +y^2 <2n^2  ⇒(2/n^2 )≤r^2 <2n^2  ⇒((√2)/n)≤r<n(√2)  ⇒A_n =∫∫_(((√2)/n)≤r<n(√2)and 0≤θ≤(π/2))    e^(−r^2 ) rdrdθ  =∫_((√2)/n) ^(n(√2))   re^(−r^2 ) dr ∫_0 ^(π/2)  dθ  =(π/2)[−(1/2)e^(−r^2 ) ]_((√2)/n) ^(n(√2))  =−(π/4){ e^(−2n^2 ) −e^(−(2/n^2 )) } ⇒  lim_(n→+∞)  A_n =(π/4) =(∫_0 ^∞ e^(−x^2 ) dx)^2   but  ∫_0 ^∞  e^(−x^2 ) dx >0 ⇒  ∫_0 ^∞  e^(−x^2 ) dx =((√π)/2) ⇒★ A =(√π)★

A=+ex2dxA=20ex2dxletAn=[1n,n[2ex2y2dxdywehaveAn=1nnex2dx.1nney2dylimn+An=(0ex2dx)2letconsiderethediffeomorphisme{x=rcosθy=rsinθ1nx<nand1ny<n2n2x2+y2<2n22n2r2<2n22nr<n2An=2nr<n2and0θπ2er2rdrdθ=2nn2rer2dr0π2dθ=π2[12er2]2nn2=π4{e2n2e2n2}limn+An=π4=(0ex2dx)2but0ex2dx>00ex2dx=π2A=π

Answered by mathmax by abdo last updated on 23/Jun/20

let use Γ function  we have ∫_(−∞) ^(+∞)  e^(−x^2 ) dx =2∫_0 ^∞  e^(−x^2 ) dx changement x^2  =t  give ∫_0 ^∞  e^(−x^2 ) dx =∫_0 ^∞  e^(−t)  (dt/(2(√t))) =(1/2)∫_0 ^∞  e^(−t)  t^(−(1/2))  dt  we know Γ(x)=∫_0 ^∞  t^(x−1)  e^(−t)  dt  (x>0) ⇒∫_0 ^∞  e^(−x^2 ) dx =(1/2)∫_0 ^∞ t^((1/2)−1)  e^(−t)  dt  =(1/2)Γ((1/2)) =((√π)/2) ⇒∫_(−∞) ^(+∞)  e^(−x^2 ) dx =(√π)

letuseΓfunctionwehave+ex2dx=20ex2dxchangementx2=tgive0ex2dx=0etdt2t=120ett12dtweknowΓ(x)=0tx1etdt(x>0)0ex2dx=120t121etdt=12Γ(12)=π2+ex2dx=π

Commented by student work last updated on 25/Jun/20

how i can Γ((1/2)) calculate?

howicanΓ(12)calculate?

Terms of Service

Privacy Policy

Contact: info@tinkutara.com