How do you integrate f(x)=(x^2+x)/((3x^2+2)(x+7)) using partial fractions?

Nov 20, 2017

$\int \frac{{x}^{2} + x}{\left(3 {x}^{2} + 2\right) \left(x + 7\right)} \mathrm{dx}$

= $\frac{23}{6} \ln | 3 {x}^{2} + 2 | - 2 \sqrt{6} {\tan}^{- 1} \left(\frac{\sqrt{3} x}{\sqrt{2}}\right) + \ln | x + 7 | + c$

Explanation:

Let $\frac{{x}^{2} + x}{\left(3 {x}^{2} + 2\right) \left(x + 7\right)} = \frac{A x + B}{3 {x}^{2} + 2} + \frac{C}{x + 7}$

Hence ${x}^{2} + x = \left(A x + B\right) \left(x + 7\right) + C \left(3 {x}^{2} + 2\right)$

or ${x}^{2} + x = A {x}^{2} + 7 A x + B x + 7 B + 3 C {x}^{2} + 2 C$

and comparing coefficients on both sides

$A + 3 C = 1$, $7 A + B = 1$ and $7 B + 2 C = 0$

the solution for these simiultaneous equations is

$A = \frac{23}{149}$, $B = - \frac{12}{149}$ and $C = \frac{42}{149}$

Hence $\int \frac{{x}^{2} + x}{\left(3 {x}^{2} + 2\right) \left(x + 7\right)} \mathrm{dx}$

= $\int \left[\frac{1}{149} \left\{\frac{23 x - 12}{3 {x}^{2} + 2} + \frac{42}{x + 7}\right\}\right] \mathrm{dx}$

= $\frac{1}{149} \int \frac{23 x - 12}{3 {x}^{2} + 2} \mathrm{dx} + \frac{42}{149} \int \frac{1}{x + 7} \mathrm{dx}$

We know $\int \frac{1}{x + 7} \mathrm{dx} = \ln | x + 7 |$ and also $\int \frac{1}{{x}^{2} + {a}^{2}} \mathrm{dx} = \frac{1}{a} {\tan}^{- 1} \left(\frac{x}{a}\right) + c$.

For $\int \frac{23 x - 12}{3 {x}^{2} + 2} \mathrm{dx}$ let us assume $u = 3 {x}^{2} + 2$ and then $\mathrm{du} = 6 x \mathrm{dx}$

hence $\int \frac{23 x - 12}{3 {x}^{2} + 2} \mathrm{dx} = \frac{23}{6} \int \frac{6 x}{3 {x}^{2} + 2} \mathrm{dx} - 4 \int \frac{1}{{x}^{2} + \frac{2}{3}} \mathrm{dx}$

= $\frac{23}{6} \int \frac{\mathrm{du}}{u} - 4 \sqrt{\frac{3}{2}} {\tan}^{- 1} \left(\frac{x}{\sqrt{\frac{2}{3}}}\right)$

= $\frac{23}{6} \ln | 3 {x}^{2} + 2 | - 2 \sqrt{6} {\tan}^{- 1} \left(\frac{\sqrt{3} x}{\sqrt{2}}\right)$

Hence $\int \frac{{x}^{2} + x}{\left(3 {x}^{2} + 2\right) \left(x + 7\right)} \mathrm{dx}$

= $\frac{23}{6} \ln | 3 {x}^{2} + 2 | - 2 \sqrt{6} {\tan}^{- 1} \left(\frac{\sqrt{3} x}{\sqrt{2}}\right) + \ln | x + 7 | + c$