Question

How do you integrate `(x) / (x^(3)-x^(2)-2x +2)` using partial fractions?

Answer 1

`=\ln|\frac{\sqrt{x^2-2}}{x-1}|+1/{\sqrt2}\ln|\frac{x-\sqrt2}{x+\sqrt2}|+C`

Explanation:

Making partial fractions as follows

`\frac{x}{x^3-x^2-2x+2}`

`=\frac{x}{(x-1)(x^2-2)}`

`=A/(x-1)+{Bx+C}/{x^2-2}`

Comparing corresponding coefficients on both sides we get

`A=-1, B=1, C=2`

Now,

`\frac{x}{x^3-x^2-2x+2}=-1/{x-1}+{x+2}/{x^2-2}`

Integrating above equation on both the sides w.r.t. `x`, we get

`\int \frac{x}{x^3-x^2-2x+2}\ dx=\int (-1/{x-1}+{x+2}/{x^2-2})\ dx`

`=-\int 1/{x-1}\ dx+\int x/{x^2-2}\ dx+2\int 1/{x^2-2}\ dx`

`=-ln|x-1|+1/2\int {d(x^2-2)}/{x^2-2}\ dx+2\int 1/{x^2-(\sqrt2)^2}\ dx`

`=-ln|x-1|+1/2\ln|x^2-2|+2\cdot 1/{2\sqrt2}\ln|\frac{x-\sqrt2}{x+\sqrt2}|+C`

`=\ln|\frac{\sqrt{x^2-2}}{x-1}|+1/{\sqrt2}\ln|\frac{x-\sqrt2}{x+\sqrt2}|+C`