Question

Evaluate $\int \frac{x^3}{(x-1)(x-2)}dx$

Answer 1

$\int \frac{x^3}{(x-1)(x-2)}dx$

$\Rightarrow \int \frac{x^3}{x^2-3x+2}dx$

$\Rightarrow \int (1+\frac{3x-2}{x^2-3x+2})dx$

$\Rightarrow \int 1dx+\int \frac{3x-2}{(x-1)(x-2)}dx$

This can be written as

$\Rightarrow \int 1dx+\int (\frac{-1}{x-1}+\frac{4}{x-2})dx$

$\Rightarrow \int 1dx+\int \frac{-1}{x-1}dx+\int \frac{4}{x-2}dx$

$\Rightarrow x-\log |x+1|+4\log |x-2|$

Therefore, $\Rightarrow \int \frac{x^3}{(x-1)(x-2)}dx=x-\log |x+1|+4\log |x-2|$