prove-that-0-x-x-ln-e-x-1-dx-n-1-1-n-3- Tinku Tara June 4, 2023 Integration 0 Comments FacebookTweetPin Question Number 33349 by caravan msup abdo. last updated on 14/Apr/18 provethat∫0∞x(x−ln(ex−1))dx=∑n=1∞1n3 Commented by math khazana by abdo last updated on 18/Apr/18 letputI=∫0∞x(x−ln(ex−1))dxI=∫0∞x(x−ln(ex(1−e−x))dx=−∫0∞xln(1−e−x)dxbutwehave11−u=∑n=0∞un⇒−ln(1−u)=∑n=0∞un+1n+1=∑n=1∞unn⇒−ln(1−e−x)=∑n=1∞e−nxnI=∫0∞(∑n=1∞e−nxn)xdx=∑n=1∞1n∫0∞xe−nxdx∫0∞xe−nxdx=nx=t∫0∞tne−tdtn=1n2∫0∞te−tdtandbyparts∫0∞te−tdt=[−te−t]0+∞+∫0∞e−tdt=[−e−t]0+∞=1⇒I=∑n=1∞1n3. Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: let-S-n-k-1-1-k-1-k-and-T-n-k-1-1-k-1-2k-1-1-calculate-lim-S-n-and-lim-T-n-n-2-prove-that-S-n-ln2-and-T-n-pi-4-converges-and-find-its-sum-Next Next post: calculate-0-lnx-x-1-4-dx- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.
![let put I = ∫_0 ^∞ x(x −ln(e^x −1))dx I = ∫_0 ^∞ x( x −ln(e^x (1−e^(−x) ))dx = −∫_0 ^∞ x ln(1−e^(−x) )dx but we have (1/(1−u))= Σ_(n=0) ^∞ u^n ⇒ −ln(1−u) =Σ_(n=0) ^∞ (u^(n+1) /(n+1)) = Σ_(n=1) ^∞ (u^n /n) ⇒ −ln(1−e^(−x) ) = Σ_(n=1) ^∞ (e^(−nx) /n) I = ∫_0 ^∞ (Σ_(n=1) ^∞ (e^(−nx) /n))x dx = Σ_(n=1) ^∞ (1/n)∫_0 ^∞ x e^(−nx) dx ∫_0 ^∞ x e^(−nx) dx =_(nx =t) ∫_0 ^∞ (t/n) e^(−t) (dt/n) =(1/n^2 ) ∫_0 ^∞ t e^(−t) dt and by parts ∫_0 ^∞ t e^(−t) dt = [−t e^(−t) ]_0 ^(+∞) +∫_0 ^∞ e^(−t) dt =[ −e^(−t) ]_0 ^(+∞) =1 ⇒ I = Σ_(n=1) ^∞ (1/n^3 ) .](https://www.tinkutara.com/question/Q33525.png)