# How do you integrate f(x)=(x^2+1)/((x^2+6)(x-4)) using partial fractions?

Mar 30, 2016

$F \left(x\right) = \frac{5}{44} \ln \left({x}^{2} + 6\right) + \frac{10}{11 \sqrt{6}} \arctan \left(\frac{x}{\sqrt{6}}\right) + \frac{17}{22} \ln | x - 4 | + c$

#### Explanation:

$f \left(x\right) = \int \frac{{x}^{2} + 1}{\left({x}^{2} + 6\right) \left(x - 4\right)} \mathrm{dx}$

Partial fraction decomposition is $\frac{A x + B}{{x}^{2} + 6} + \frac{C}{x - 4}$

$\left(A x + B\right) \left(x - 4\right) + C \left({x}^{2} + 6\right) = {x}^{2} + 0 x + 1$
From ${x}^{2}$ terms: $A + C = 1$
From ${x}^{1}$ terms: $B - 4 A = 0$
From ${x}^{0}$ terms: $6 C - 4 B = 1$
Simultaneous solving yields $A = \frac{5}{22}$, $B = \frac{20}{22}$ and $C = \frac{17}{22}$
Alternatively, use cover up rule where possible.

Thus the partial fraction decomposition is
$\frac{5 x + 20}{22 \left({x}^{2} + 6\right)} + \frac{17}{22 \left(x - 4\right)}$

Thus $f \left(x\right) = \int \frac{5 x + 20}{22 \left({x}^{2} + 6\right)} + \frac{17}{22 \left(x - 4\right)} \mathrm{dx}$

Algebraic manipulation yields:
$f \left(x\right) = \frac{5}{44} \int \frac{2 x}{{x}^{2} + 6} \mathrm{dx} + \frac{10}{11} \int \frac{1}{{x}^{2} + 6} \mathrm{dx} + \frac{17}{22} \int \frac{1}{x - 4} \mathrm{dx}$

Integrating yields:
$F \left(x\right) = \frac{5}{44} \ln \left({x}^{2} + 6\right) + \frac{10}{11 \sqrt{6}} \arctan \left(\frac{x}{\sqrt{6}}\right) + \frac{17}{22} \ln | x - 4 | + c$