Messy Integral of polynomial over polynomial

Question

How do I solve the integral

$$\int \frac{(x^3+6x^2+3x+16)}{(x^3+4x)} dx? $$

This integral gets very messy. Can I get a step by step break down of how to solve it?

Answer 1

1) Factor the denominator

2) Polynomial + partial fractions

3) Integrate term by term.

Answer 2

Do polynomial long division (I will not do that here) to get $$ \int \left(\frac{2x-1}{x^2+4}+\frac{4}{x}+1\right)dx \\ 2\int\frac{x}{x^2+4}dx-\int\frac{1}{x^2+4}dx+4\int\frac{1}{x}dx+\int dx. $$ The last two integrals are easy to integrate. The first two integrals need substitutions. For the first integral, use $s=x^2+4$ and $t=\frac{x}{2}$ for the second integral (after factoring out $4$ from the denominator). After integrating, we get $$ \ln(s)-\frac{1}{2}\tan^{-1}(t)+4\ln(x)+x+c $$ for a constant $c$. Replace $s$ and $t$ with your former substitutions.