# How do you integrate int (x-3x^2)/((x-6)(x-2)(x-5))  using partial fractions?

Jul 4, 2016

$\int \frac{x - 3 {x}^{2}}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)} \mathrm{dx}$

= $- \frac{121}{8} \ln \left(x - 6\right) + \frac{5}{8} \ln \left(x - 2\right) + \frac{35}{4} \ln \left(x - 5\right) + c$

#### Explanation:

Let us first convert $\frac{x - 3 {x}^{2}}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)}$ into partial fractions.

$\frac{x - 3 {x}^{2}}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)} \Leftrightarrow \frac{A}{x - 6} + \frac{B}{x - 2} + \frac{C}{x - 5}$

$\frac{x - 3 {x}^{2}}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)} \Leftrightarrow \frac{A \left(x - 2\right) \left(x - 5\right) + B \left(x - 6\right) \left(x - 5\right) + C \left(x - 6\right) \left(x - 2\right)}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)}$

= $\frac{A \left({x}^{2} - 7 x + 10\right) + B \left({x}^{2} - 11 x + 30\right) + C \left({x}^{2} - 8 x + 12\right)}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)}$

= $\frac{{x}^{2} \left(A + B + C\right) - x \left(7 A + 11 B + 8 C\right) + \left(10 A + 30 B + 12 C\right)}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)}$

Hence $A + B + C = - 3$, $7 A + 11 B + 8 C = - 1$ and $10 A + 30 B + 12 C = 0$

Subtracting $7$ times of first equation from second and $10$ times first from third equation, we get

$4 B + C = 20$ and $20 B + C = 30$ which gives us $B = \frac{5}{8}$ and $C = \frac{140}{8} = \frac{35}{4}$ and putting these in $A + B + C = - 3$, we get $A = - \frac{121}{8}$

Hence, $\frac{x - 3 {x}^{2}}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)} = - \frac{121}{8 \left(x - 6\right)} + \frac{5}{8 \left(x - 2\right)} + \frac{35}{4 \left(x - 5\right)}$

and $\int \frac{x - 3 {x}^{2}}{\left(x - 6\right) \left(x - 2\right) \left(x - 5\right)} \mathrm{dx} = \int \left[- \frac{121}{8 \left(x - 6\right)} + \frac{5}{8 \left(x - 2\right)} + \frac{35}{4 \left(x - 5\right)}\right] \mathrm{dx}$

= $- \frac{121}{8} \ln \left(x - 6\right) + \frac{5}{8} \ln \left(x - 2\right) + \frac{35}{4} \ln \left(x - 5\right) + c$