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Question Number 146577 by bekzodjumayev last updated on 14/Jul/21

Commented by bekzodjumayev last updated on 14/Jul/21

$$Help$$$$$$

Answered by Ar Brandon last updated on 14/Jul/21

$$\mathrm{I}=\int \frac{\mathrm{dx}}{\mathrm{x}(\mathrm{x}+1)(\mathrm{x}+2)\cdot \cdot \cdot (\mathrm{x}+\mathrm{m})}$$$$=\int (\frac{1}{\mathrm{m}!}\cdot \frac{1}{\mathrm{x}}-\frac{1}{(\mathrm{m}-1)!}\cdot \frac{1}{\mathrm{x}+1}+\frac{1}{2(\mathrm{m}-2)!}\cdot \frac{1}{\mathrm{x}+2}$$$$(-\frac{1}{3!(\mathrm{m}-3)!}\cdot \frac{1}{\mathrm{x}+3}+\cdot \cdot \cdot +{(-1)}^{\mathrm{m}}\frac{1}{\mathrm{m}!}\cdot \frac{1}{\mathrm{x}+\mathrm{m}})\mathrm{dx}$$$$=\int \underset{\mathrm{k}=0}{\sum ^{\mathrm{m}}}{(-1)}^{\mathrm{k}}\frac{1}{\mathrm{k}!(\mathrm{m}-\mathrm{k})!}\cdot \frac{1}{\mathrm{x}+\mathrm{k}}\mathrm{dx}=\underset{\mathrm{k}=0}{\sum ^{\mathrm{m}}}{(-1)}^{\mathrm{k}}\frac{\ln \mid \mathrm{x}+\mathrm{k}\mid }{\mathrm{k}!(\mathrm{m}-\mathrm{k})!}+\mathrm{C}$$

Commented by bekzodjumayev last updated on 14/Jul/21

$$Thankyou$$

Answered by mathmax by abdo last updated on 14/Jul/21

$$\mathrm{\Psi }=\int \frac{\mathrm{dx}}{\mathrm{x}(\mathrm{x}+1)(\mathrm{x}+2).....(\mathrm{x}+\mathrm{m})}$$$$\mathrm{let}\mathrm{decompose}\mathrm{F}\left(\mathrm{x}\right)=\frac{1}{\mathrm{x}(\mathrm{x}+1)....(\mathrm{x}+\mathrm{m})}=\sum _{\mathrm{k}=0}^{\mathrm{m}}\frac{{\mathrm{a}}_{\mathrm{k}}}{\mathrm{x}+\mathrm{k}}$$$${\mathrm{a}}_{\mathrm{k}}={\lim }_{\mathrm{x}\to -\mathrm{k}}(\mathrm{x}+\mathrm{k})\mathrm{F}\left(\mathrm{x}\right)$$$$={\lim }_{\mathrm{x}\to -\mathrm{k}}(\mathrm{x}+\mathrm{k})\times \frac{1}{\mathrm{x}(\mathrm{x}+1).....(\mathrm{x}+\mathrm{k}-1)(\mathrm{x}+\mathrm{k})(\mathrm{x}+\mathrm{k}+1)....(\mathrm{x}+\mathrm{m})}$$$$=\frac{1}{(-\mathrm{k})(-\mathrm{k}+1).....(-1)\times 1.2.....(-\mathrm{k}+\mathrm{m})}$$$$=\frac{1}{{(-1)}^{\mathrm{k}}\mathrm{k}!(\mathrm{m}-\mathrm{k})!}=\frac{{(-1)}^{\mathrm{k}}}{\mathrm{m}!}\times \frac{\mathrm{m}!}{\mathrm{k}!(\mathrm{m}-\mathrm{k})!}=\frac{{(-1)}^{\mathrm{k}}}{\mathrm{m}!}{\mathrm{C}}_{\mathrm{m}}^{\mathrm{k}}\Rightarrow$$$$\mathrm{F}\left(\mathrm{x}\right)=\frac{1}{\mathrm{m}!}\sum _{\mathrm{k}=0}^{\mathrm{m}}\frac{{(-1)}^{\mathrm{k}}{\mathrm{C}}_{\mathrm{m}}^{\mathrm{k}}}{\mathrm{x}+\mathrm{k}}\Rightarrow$$$$\mathrm{\Psi }=\int \mathrm{F}\left(\mathrm{x}\right)\mathrm{dx}=\frac{1}{\mathrm{m}!}\sum _{\mathrm{k}=0}^{\mathrm{m}}{(-1)}^{\mathrm{k}}{\mathrm{C}}_{\mathrm{m}}^{\mathrm{k}}\log \mid \mathrm{x}+\mathrm{k}\mid +\mathrm{C}$$$$$$

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