\textbf{Partial fraction decomposition}

How do you solve \[\int\frac{P(x)}{Q(x)}\mathrm{d}x\] where \[\ \deg{P(x)} < \deg{Q(x)}\]? 


First, as a consequence of the fundamental theorem of algebra, any real polynomial can be factored into linear and quadtratic terms. 
We will do that with \[Q(x)\]:

,,\qquad Q(x) = (x-a_1)^{A_1}(x-a_2)^{A_2}\dots(x - a_m)^{A_m}(x^2 + b_1x + c_1)^{B_1}(x^2 + b_2 x + c_2)^{B_2}\dots(x^2 + b_n x + c_n)^{B_n}

Now  employ the partial fraction decomposition:

,,\qquad\frac{P(x)}{Q(x)} = \sum_{i=1}^m\sum_{j=1}^{A_i}\frac{a_{ij}}{(x-a_i)^j} + \sum_{i=1}^n\sum_{j=1}^{B_i}\frac{b_{ij}x + c_{ij}}{(x^2 + b_ix +)^j}

Then just use the linearity property of the integral.

For example:

,,\qquad \int\frac{x^2 + 1}{x^5 - 3x^4 + x^3 + 7x^2 - 6x - 8}\mathrm{d}x = \int\frac{x^2 + 1}{(x-2)(x +1)^2(x^2-3x + 4)}\mathrm{d}x = \int \frac{a_{11}}{x-2}\mathrm{d}x + \int\frac{a_{21}}{x+1}\mathrm{d}x + \int\frac{a_{22}}{(x+1)^2}\mathrm{d}x + \int\frac{b_{11} x + c_{11}}{x^2 - 3x + 4} \mathrm{d}x


The constants \[a_{ij},b_{ij}, c_{ij}\in\mathbb{R}\] have to be found e.g. using the \href{https://en.wikipedia.org/wiki/Heaviside_cover-up_method}{Heaviside cover up method}.