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calculate-0-6-e-x-x-1-e-x-dx-




Question Number 37280 by abdo.msup.com last updated on 11/Jun/18
calculate  ∫_0 ^6    (e^(x−[x]) /(1+e^x ))dx .
calculate06ex[x]1+exdx.
Commented by prof Abdo imad last updated on 16/Jun/18
I = Σ_(k =0) ^5  ∫_k ^(k+1)    (e^(x−k) /(1+e^x ))dx  =Σ_(k=0) ^5  e^(−k)   ∫_k ^(k+1)    (e^x /(1+e^x ))dx  =Σ_(k=0) ^5  e^(−k) [ln(1+e^x )]_k ^(k+1)   =Σ_(k=0) ^5 e^(−k)  {ln(1+e^(k+1) ) −ln(1 +e^k )}  =Σ_(k=0) ^5  e^(−k)  ln(((1+e^(k+1) )/(1+e^k )))  =ln(((1+e)/2))  +e^(−1) ln( ((1+e^2 )/(1+e))) +e^(−2) ln(((1+e^3 )/(1+e^2 )))  +e^(−3) ln(((1+e^4 )/(1+e^3 ))) +e^(−4) ln( ((1+e^5 )/(1 +e^4 ))) +e^(−5) ln(((1+e^6 )/(1+e^5 ))) .
I=k=05kk+1exk1+exdx=k=05ekkk+1ex1+exdx=k=05ek[ln(1+ex)]kk+1=k=05ek{ln(1+ek+1)ln(1+ek)}=k=05ekln(1+ek+11+ek)=ln(1+e2)+e1ln(1+e21+e)+e2ln(1+e31+e2)+e3ln(1+e41+e3)+e4ln(1+e51+e4)+e5ln(1+e61+e5).

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