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Question Number 146035 by KONE last updated on 10/Jul/21

help me please ∫((ln(x+1))/x)dx=??

$${help}\:{me}\:{please} \\ $$$$\int\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}=?? \\ $$$$ \\ $$

Answered by KONE last updated on 10/Jul/21

please

$${please} \\ $$

Answered by puissant last updated on 10/Jul/21

(1/(1+x))=Σ_(n=0) ^∞ (−1)^n x^n ⇒ln(1+x)=∫Σ_(n=0) ^∞ (−1)^n x^n dx ⇒ln(1+x)=Σ_(n=0) ^∞ (−1)^n (x^(n+1) /(n+1))+c x=0 ⇒ ln(1+x)=Σ_(n=0) ^∞ (−1)^n (x^(n+1) /(n+1)) ⇒((ln(1+x))/x)=Σ_(n=0) ^∞ (−1)^n (x^n /(n+1)) ⇒∫((ln(1+x))/x)dx=Σ_(n=0) ^∞ (−1)^n (1/(n+1))∫x^n dx ⇒ I=Σ_(n=0) ^∞ (−1)^n (x^(n+1) /((n+1)^2 ))+k..

$$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}}=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \\ $$$$\Rightarrow\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)=\int\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \mathrm{dx} \\ $$$$\Rightarrow\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\mathrm{n}+\mathrm{1}}+\mathrm{c} \\ $$$$\mathrm{x}=\mathrm{0}\:\Rightarrow\:\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\mathrm{n}+\mathrm{1}} \\ $$$$\Rightarrow\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}}=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{n}+\mathrm{1}} \\ $$$$\Rightarrow\int\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx}=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}}\int\mathrm{x}^{\mathrm{n}} \mathrm{dx} \\ $$$$\Rightarrow\:\mathrm{I}=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{n}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{2}} }+\mathrm{k}.. \\ $$

Commented by KONE last updated on 12/Jul/21

thanks... svp est possible de faire sans utiliser le DL?

$${thanks}... \\ $$$${svp}\:{est}\:{possible}\:{de}\:{faire}\:{sans}\:{utiliser}\:{le}\:{DL}? \\ $$

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