Partial Fractions Integration Question

Question

$$\int\frac{x^5+x-1}{x^3 +1} dx$$

Have tried everything ... polynomial long division, partial fractions, trig substitution etc...

Not for an assignment, so if a complete solution could be provided that'd be much appreaciated

Answer 1

Hint:

$$\frac{x^5+x-1}{x^3 +1} = \frac{(x^3+x^2-1)(x^2-x+1)}{(x+1)(x^2-x+1)}$$

Answer 2

First divide the denominator into the numerator to get

$$\frac{x^5+x-1}{x^3+1}=x^2 -\frac{x^2-x+1}{x^3+1}$$

Now we factor the denominator, $$x^3+1=(x+1)(x^2-x+1)$$

Note that we have cancellation so we get $$\int x^2-\frac{1}{x+1} dx=\frac{x^3}{3}-\ln |x+1|+C$$