Integrate (1+x^2)/(1+x^3)

Question

I just can't manage to integrate this expression, I need help

Answer 1

I was working on this problem also. With all due respect, I am not sure I understand Sam Z.'s answer.

Here how I would do it...and it is straight forward, but messy.

First break the integrand into 2 pieces [1/(1+x³)] and [x²/(1+x³)]

The 2nd term integrates as a ln (1+x³).

Decompose the 2nd term into partial fractions:

1/(1+x³) = A/(1+x) + [(Bx+C)/(1-x+x²)

When I did this I got A=1/3, B=-1/3 and C = 2/3

The term in A integrates as a ln (1+x)

The 2nd term again integrates in 2 parts:

You get a term in ln(1-x+x²) and then a term in arctan by completing the square in the denominator.

Messy as I said...and if you can get clear what Sam Z.was writing, maybe his solution is easier and better.

Answer 2

(1+x^2)/(1+x^3)

" " " * ( " " " )^-1

+/

(1+(x^3)/3)/(1+(x^4)/4)

" " " * " " "^-1

By adding 1 to the power of the denominator; It'll be the same.