Решите интеграл (x+2/x^2(x-1)) *dx. Created by Shalshek. matematika-ru

x^2(x-1)) *dx...

Shalshek @Shalshek

September 2021 1 39 Report

Решите интеграл (x+2/x^2(x-1)) *dx

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$\int\limits^3_2 \frac{x+2}{x^2(x-1)} \, dx =Q\\\\ \frac{x+2}{x^2(x-1)}=\frac{A}{x^2}+\frac{B}{x}+\frac{C}{x-1} \\\\x+2=A(x-1)+Bx(x-1)+Cx^2\\\\x=0:\; \; A= \frac{2}{-1}=-2\\\\x=1:\; \; C= \frac{1+2}{1}=3\\\\x^2|\; 0=B+C\; \' ,\; \; B=-C=-3\\\\\int\limits \frac{x+2}{x^2(x-1)}\, dx=-2\, \int\limits \frac{dx}{x^2}-3\, \int\limits \frac{dx}{x} +3\, \int\limits \frac{dx}{x-1}=\\\\=-2\cdot \frac{-1}{x}-3\, ln|x|+3\, ln|x-1|+C=\frac{2}{x}+3\, ln| \frac{x-1}{x} |+C$

$Q= (\frac{2}{x}+2\, ln| \frac{x-1}{x}|)\Big |_2^3= \frac{2}{3}+2\, ln \frac{2}{3}-1-2\, ln \frac{1}{2}=- \frac{1}{3}+2\, ln \frac{4}{3}$

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