Mathematics

Evaluate $$\displaystyle \int {\dfrac {(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}}dx$$


SOLUTION
$$I=\displaystyle \int {\dfrac {(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}}dx\\\implies \dfrac {(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}=\dfrac A{x-4}+\dfrac B{x-5}+\dfrac C{x-6}\\\implies (x-1)(x-2)(x-3)=A(x-5)(x-6)+B(x-4)(x-6)+C(x-4)(x-5)$$Put $$x=5$$$$4\times3\times2=B(1)(-1)\\\implies B=-24$$Put $$x=6$$$$5\times 4\times 3=C(2)(1)\\\implies C=30$$Put $$x=4$$$$3\times 2\times 1=A(-1)(-2)\\\implies A=3$$Therefore, $$I=\int {\left(\dfrac 3{x-4}-\dfrac {24}{x-5}+\dfrac {30}{x-6}\right)}dx\\\implies I=3\ln{|x-4|}-24\ln{|x-5|}+30\ln{|x-6|}+C$$
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Subjective Medium Published on 17th 09, 2020
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