How do you integrate int (1-x^2)/((x-9)(x+6)(x-2))  using partial fractions?

Mar 31, 2018

$\int \frac{1 - {x}^{2}}{\left(x - 9\right) \left(x + 6\right) \left(x - 2\right)} \mathrm{dx}$

$= - \frac{16}{21} \ln \left\mid x - 9 \right\mid - \frac{7}{24} \ln \left\mid x + 6 \right\mid + \frac{3}{56} \ln \left\mid x - 2 \right\mid + C$

Explanation:

Conveniently, the denominator in this problem is already factored.

To apply partial fraction decomposition, we want to express

$\int \frac{1 - {x}^{2}}{\left(x - 9\right) \left(x + 6\right) \left(x - 2\right)} \mathrm{dx}$

in the form

$\int \left(\frac{A}{x - 9} + \frac{B}{x + 6} + \frac{C}{x - 2}\right) \mathrm{dx}$

So we can equate

$\frac{1 - {x}^{2}}{\left(x - 9\right) \left(x + 6\right) \left(x - 2\right)} = \frac{A}{x - 9} + \frac{B}{x + 6} + \frac{C}{x - 2}$

Multiplying both sides by the denominator of the left, we get:

$1 - {x}^{2} = A \left(x + 6\right) \left(x - 2\right) + B \left(x - 9\right) \left(x - 2\right) + C \left(x - 9\right) \left(x + 6\right)$

Now let's expand the parentheses and group similar terms:

$1 - {x}^{2} = A \left({x}^{2} + 4 x - 12\right) + B \left({x}^{2} - 11 x + 18\right) + C \left({x}^{2} - 3 x - 54\right)$

$1 - {x}^{2} = \left(A + B + C\right) {x}^{2} + \left(4 A - 11 B - 3 C\right) x + \left(- 12 A + 18 B - 54 C\right)$

Now we can compare the coefficients on both sides, and write a system of three equations:

$A + B + C = - 1$
$4 A - 11 B - 3 C = 0$
$- 12 A + 18 B - 54 C = 1$

Solving this system (whichever way you choose) gives:

$A = - \frac{16}{21}$

$B = - \frac{7}{24}$

$C = \frac{3}{56}$

So now we can rewrite the integral as:

$\int \left(- \frac{16}{21} \frac{1}{x - 9} - \frac{7}{24} \frac{1}{x + 6} + \frac{3}{56} \frac{1}{x - 2}\right) \mathrm{dx}$

$\Rightarrow - \frac{16}{21} \int \frac{1}{x - 9} \mathrm{dx} - \frac{7}{24} \int \frac{1}{x + 6} \mathrm{dx} + \frac{3}{56} \int \frac{1}{x - 2} \mathrm{dx}$

Integrating, we get:

$- \frac{16}{21} \ln \left\mid x - 9 \right\mid - \frac{7}{24} \ln \left\mid x + 6 \right\mid + \frac{3}{56} \ln \left\mid x - 2 \right\mid + C$