# How do you integrate int (1-2x^2)/((x+1)(x-6)(x-7))  using partial fractions?

Oct 14, 2016

$\int \frac{1 - 2 {x}^{2}}{\left(x + 1\right) \left(x - 6\right) \left(x - 7\right)} \mathrm{dx}$

$= - \frac{1}{56} \ln \left\mid x + 1 \right\mid + \frac{71}{7} \ln \left\mid x - 6 \right\mid - \frac{97}{8} \ln \left\mid x - 7 \right\mid + C$

#### Explanation:

$\int \frac{1 - 2 {x}^{2}}{\left(x + 1\right) \left(x - 6\right) \left(x - 7\right)} \mathrm{dx}$

$= \int \left(- \frac{1}{56} \left(\frac{1}{x + 1}\right) + \frac{71}{7} \left(\frac{1}{x - 6}\right) - \frac{97}{8} \left(\frac{1}{x - 7}\right)\right) \mathrm{dx}$

$= - \frac{1}{56} \ln \left\mid x + 1 \right\mid + \frac{71}{7} \ln \left\mid x - 6 \right\mid - \frac{97}{8} \ln \left\mid x - 7 \right\mid + C$

$\textcolor{w h i t e}{}$
Where did those coefficients come from?

$\frac{1 - 2 {x}^{2}}{\left(x + 1\right) \left(x - 6\right) \left(x - 7\right)} = \frac{a}{x + 1} + \frac{b}{x - 6} + \frac{c}{x - 7}$

We can calculate $a , b , c$ using Heaviside's cover up method:

$a = \frac{1 - 2 {\left(\textcolor{b l u e}{- 1}\right)}^{2}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(\left(\textcolor{b l u e}{- 1}\right) + 1\right)}}} \left(\left(\textcolor{b l u e}{- 1}\right) - 6\right) \left(\left(\textcolor{b l u e}{- 1}\right) - 7\right)} = \frac{- 1}{\left(- 7\right) \left(- 8\right)} = - \frac{1}{56}$

$b = \frac{1 - 2 {\left(\textcolor{b l u e}{6}\right)}^{2}}{\left(\left(\textcolor{b l u e}{6}\right) + 1\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(\left(\textcolor{b l u e}{6}\right) - 6\right)}}} \left(\left(\textcolor{b l u e}{6}\right) - 7\right)} = \frac{- 71}{\left(7\right) \left(- 1\right)} = \frac{71}{7}$

$c = \frac{1 - 2 {\left(\textcolor{b l u e}{7}\right)}^{2}}{\left(\left(\textcolor{b l u e}{7}\right) + 1\right) \left(\left(\textcolor{b l u e}{7}\right) - 6\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(\left(\textcolor{b l u e}{7}\right) - 7\right)}}}} = \frac{- 97}{\left(8\right) \left(1\right)} = - \frac{97}{8}$

Oct 14, 2016