Question

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

Answer 1

Consider, $I=\int \frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}dx$

$I=\int 1+\frac{9x^2-63x+114}{(x-4)(x-5)(x-6)}dx$

$let\frac{9x^2-63x+114}{(x-4)(x-5)(x-6)}=\frac{A}{x-4}+\frac{B}{x-5}+\frac{C}{x-6}$

$OnsolvingtheaboveequationwegetA=3,B=-24,C=30$

$Nowtheintegrationreducesto;$

$\Rightarrow$ $\int 1dx+\int \frac{3}{x-4}dx+\int \frac{-24}{x-5}dx+\int \frac{30}{x-6}dx$

$=x+3\ln (x-4)-24\ln (x-5)+30\ln (x-6)$