Evaluate the integral $\int \frac{dx}{x^3 + 2x^2 + 2x}$ of a rational function

Question

Evaluate $$\int \frac{dx}{x^3 + 2x^2 + 2x}.$$

I have no idea how to approach this. I know how to solve rational functions with numerator as highest degree polynomial using division and remainder.

How do I go about this type of question?

I am more interested in the approach and general procedure rather than the answer itself, so that I can answer questions like these in the future.

Answer 1

Hint Using the factorization of the denominator, we can rewrite the integrand as $$\frac{1}{x^3 + 2x^2 + 2x} = \frac{A x + B}{x^2 + 2 x + 2} + \frac{C}{x}$$ for some unique constants $A, B, C$. To find those constants, cross-multiply to clear denominators, distribute, and then compare like terms in $x$. This is the so-called partial fractions decomposition of the given rational function.

Additional hint When integrating the first summand above, substitute $u = x + 1$ so that the denominator becomes a sum $u^2 + 1$ of squares.

Answer 2

Hint:

First, you must re-write the integrand using partial fractions decomposition:

$\begin{align*} \frac{1}{x^3+2x^2+2x}&=\frac{\frac{1}{2}}{x}+\frac{-\frac{1}{2}x-1}{x^2+2x+2}\\ &=\frac{1}{2}\frac{1}{x}-\frac{1}{2}\frac{(x+1)+1}{(x+1)^2+1} \end{align*}$

Answer 3

Extract $x$ out of denominator and you can have

$$\frac{1}{x^3+2x^2+2x} = \frac{1}{2}\left(\frac{1}{x} - \frac{x+2}{x^2+2x+2}\right)$$

Now you can integrate. If you still have problem on the second term, the hint is use substitution.