Монгол Бодлогын Сан

Заавар:$$\dfrac{5x}{(x-1)^3}=\dfrac{A}{(x-1)^3}+\dfrac{B}{(x-1)^2}+\dfrac{C}{x-1}$$ байх $A$, $B$, $C$ тоонуудыг ол.

Бодолт:$\dfrac{5x}{(x-1)^3}=\dfrac{A}{(x-1)^3}+\dfrac{B}{(x-1)^2}+\dfrac{C}{x-1}$ бол $$A+B(x-1)+C(x-1)^2=5x$$ буюу $$Cx^3+(B-2C)x^2+A-B+C=5x$$ тул $$\left\{\begin{array}{c} C=0\\ B-2C=5\\ A-B+C=0 \end{array}\right.\Rightarrow\left\{\begin{array}{l} A=5\\ B=5\\ C=0 \end{array}\right.$$ Иймд \begin{align*} \text{Инт.}&=\int\dfrac{5x}{(x-1)^3}=\int\left(\dfrac{5}{(x-1)^3}+\dfrac{5}{(x-1)^2}\right)dx\\ &=5\int\dfrac{dx}{(x-1)^3}+5\int\dfrac{dx}{(x-1)^2}\\ &=5\cdot\dfrac{(x-1)^{-2}}{-2}-\dfrac{5}{x-1}+C\\ &=-\dfrac{5}{2(x-1)^2}-\dfrac{5}{x-1}+C \end{align*}