Integration of elementary rational fractions

$$\int{\frac{Mx+N}{x^2+px+q}}dx$$

When all the coefficients are given, it can be easily solved using substitution but I have no idea how to find a solution in a standard form.

What are the first steps?

Solution 1:

• Since the derivative of $x^{2}+px+q$ is $2x+p$, we first try to make the numerator $Mx+N$ as $2x+p$. To achieve this, write the numerator as $$\frac{M}{2} \cdot \Bigl(2x+\frac{2N}{M}\Bigr)=\frac{M}{2}\cdot\left(2x+p +\frac{2N}{M}-p\right)=\frac{M}{2}\cdot (2x+p) + \frac{M}{2} \cdot\left(\frac{2N}{M}-p\right)$$

• Then your integral becomes $$\frac{M}{2}\cdot \left[\,\int \frac{2x+p}{x^{2}+px+q} +\left(\frac{2N}{M}-p\right) \cdot \int \frac{1}{x^{2}+px+q}\,\right]$$

• Integral $$I_{1}=\int\frac{2x+p}{x^2+px+q} = \ln\,(x^{2}+px+q)+C$$ whereas the integral $$I_{2}=\int\frac{1}{x^{2}+px+q}$$ can be evaluated by noting that $$x^{2}+px+q = \left(x+\frac{p}{2}\right)^{2}-\left\{\sqrt{q-\frac{p^2}{4}}\right\}^{2}$$ and using standard substitutions.