<s>[INST] Partial Fractions Integration 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
 [/INST] 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
$$\int x^2-\frac{1}{x+1} dx=\frac{x^3}{3}-\ln |x+1|+C$$
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