Question Number 37280 by abdo.msup.com last updated on 11/Jun/18
![calculate ∫_0 ^6 (e^(x−[x]) /(1+e^x ))dx .](https://www.tinkutara.com/question/Q37280.png)
$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{6}} \:\:\:\frac{{e}^{{x}−\left[{x}\right]} }{\mathrm{1}+{e}^{{x}} }{dx}\:. \\ $$
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 ))) .](https://www.tinkutara.com/question/Q37673.png)
$${I}\:=\:\sum_{{k}\:=\mathrm{0}} ^{\mathrm{5}} \:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\:\frac{{e}^{{x}−{k}} }{\mathrm{1}+{e}^{{x}} }{dx} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{5}} \:{e}^{−{k}} \:\:\int_{{k}} ^{{k}+\mathrm{1}} \:\:\:\frac{{e}^{{x}} }{\mathrm{1}+{e}^{{x}} }{dx} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{5}} \:{e}^{−{k}} \left[{ln}\left(\mathrm{1}+{e}^{{x}} \right)\right]_{{k}} ^{{k}+\mathrm{1}} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{5}} {e}^{−{k}} \:\left\{{ln}\left(\mathrm{1}+{e}^{{k}+\mathrm{1}} \right)\:−{ln}\left(\mathrm{1}\:+{e}^{{k}} \right)\right\} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{5}} \:{e}^{−{k}} \:{ln}\left(\frac{\mathrm{1}+{e}^{{k}+\mathrm{1}} }{\mathrm{1}+{e}^{{k}} }\right) \\ $$$$={ln}\left(\frac{\mathrm{1}+{e}}{\mathrm{2}}\right)\:\:+{e}^{−\mathrm{1}} {ln}\left(\:\frac{\mathrm{1}+{e}^{\mathrm{2}} }{\mathrm{1}+{e}}\right)\:+{e}^{−\mathrm{2}} {ln}\left(\frac{\mathrm{1}+{e}^{\mathrm{3}} }{\mathrm{1}+{e}^{\mathrm{2}} }\right) \\ $$$$+{e}^{−\mathrm{3}} {ln}\left(\frac{\mathrm{1}+{e}^{\mathrm{4}} }{\mathrm{1}+{e}^{\mathrm{3}} }\right)\:+{e}^{−\mathrm{4}} {ln}\left(\:\frac{\mathrm{1}+{e}^{\mathrm{5}} }{\mathrm{1}\:+{e}^{\mathrm{4}} }\right)\:+{e}^{−\mathrm{5}} {ln}\left(\frac{\mathrm{1}+{e}^{\mathrm{6}} }{\mathrm{1}+{e}^{\mathrm{5}} }\right)\:. \\ $$$$ \\ $$