As you have written now, you don't even have an integral.
Careful with parentheses:
int([(2x^6 + 1)/(x^6)]*(1 + x^2)dx)
Is that what you're trying to compute?
If so:
Expand (x^2 + 1)*(2x^6 + 1) = 2x^8 + 2x^6 + x^2 + 1
And thus, (2x^8 + 2x^6 + x^2 + 1)/x^6
Simplify: 2x^2 + 1/x^4 + 1/x^6 + 2
Integrate: (2x^3)/3 - 1/(3x^3) - 1/(5x^5) + 2x + C
I think he's actually trying to integrate:
INT (2x^6 + 1)/[(x^6)(1 + x^2)] dx ... the x^6 and 1 + x^2 both being on the denominator.
Most of the problems I have helped him on so far are actually very long and complicated. He has yet to integrate something as simple as some product of polynomials divided by x^6. So you are very correct in that he needs to be careful with parenthesis.
7. INT (2X^6 + 1)/[(x^6)(1 + x^2)] dx
This one is pretty interesting. I won't go through the entire process of solving it, but I will set up the final integration:
First, add and subtract 1 from the numerator
INT (2x^6 + 2 - 1)/[(x^6)(1 + x^2)] dx
INT 2(x^6 + 1)/[(x^6)(1 + x^2)] dx - INT 1/[(x^6)(1 + x^2)] dx
This numerator is a sum of cubes, which can be factored:
2*INT [(x^2 + 1)(x^4 - x^2 + 1)]/[(x^6)(1 + x^2)] dx
2*INT (x^4 - x^2 + 1)/x^6 dx
2*INT (1/x^2 - 1/x^4 + 1/x^6) dx
Add and subtract x^2 from this integration:
INT (1 + x^2 - x^2)/[(x^6)(1 + x^2)] dx
INT (1 + x^2)/[(x^6)(1 + x^2)] dx - INT x^2/[(x^6)(1 + x^2)] dx
INT 1/x^6 - INT 1/[(x^4)(1 + x^2)] dx
Add and subtract x^2 from this integration:
INT (1 + x^2 - x^2)/[(x^4)(1 + x^2)] dx
INT (1 + x^2)/[(x^4)(1 + x^2)] dx - INT x^2/[(x^4)(1 + x^2)] dx
INT 1/x^4 dx - INT 1/[(x^2)(1 + x^2)] dx
Add and subtract x^2 from this integration:
INT (1 + x^2 - x^2)/[(x^2)(1 + x^2)] dx
INT (1 + x^2)/[(x^2)(1 + x^2)] dx - INT x^2/[(x^2)(1 + x^2)] dx
INT 1/x^2 dx - INT 1/(1 + x^2) dx
Combine the entire integration into one (be careful with negative signs) and integrate. It will be much easier to do at this point.