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Evaluate-0-x-1-2020-x-2-2022-dx-




Question Number 166756 by HongKing last updated on 27/Feb/22
Evaluate:  ∫_0 ^( ∞)  (((x + 1)^(2020) )/((x + 2)^(2022) )) dx = ?
Evaluate:0(x+1)2020(x+2)2022dx=?
Answered by Mathspace last updated on 27/Feb/22
x+2=t ⇒I=∫_2 ^∞ (((t−1)^(2020) )/t^(2022) )dt  =∫_2 ^∞  ((Σ_(k=0) ^(2020) C_(2020) ^k t^k (−1)^(2020−k) )/t^(2022) )dt  =Σ_(k=0) ^(2020) (−1)^k  C_(2020) ^k  t^(k−2022) dt  =Σ_(k=0) ^(2020) (−1)^k  C_(2020) ^k [(1/(k−2021))t^(k−2022) ]_2 ^∞   =−Σ_(k=0) ^(2020) (((−1)^k C_(2020) ^k )/(k−2021)) 2^(k−2022)
x+2=tI=2(t1)2020t2022dt=2k=02020C2020ktk(1)2020kt2022dt=k=02020(1)kC2020ktk2022dt=k=02020(1)kC2020k[1k2021tk2022]2=k=02020(1)kC2020kk20212k2022
Answered by mathsmine last updated on 27/Feb/22
=∫_0 ^∞ (((x+1)/(x+2)))^(2020) .((1/(x+2)))^2 dx  u=((x+1)/(x+2))=1−(1/((x+2)))⇒du=(dx/((x+2)^2 ))  =∫_(1/2) ^1 u^(2020) du=(1/(2021))(1−(1/2^(2021) ))
=0(x+1x+2)2020.(1x+2)2dxu=x+1x+2=11(x+2)du=dx(x+2)2=121u2020du=12021(1122021)

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