# Evaluate integral of a polynomial over another polynomial (complicated substitution)

1. Dec 1, 2011

### skyturnred

1. The problem statement, all variables and given/known data

$\int^{√x}_{1}$$\frac{t^{3}+t-1}{t^{2}(t^{2}+1)}$ dt

2. Relevant equations

3. The attempt at a solution

So I first start by expanding the bottom part of the fraction to t$^{4}$+t$^{2}$, and letting u equal to that. Then du=4t$^{3}$+2t dt. I move the common multiple of 2 over to the other side so that it is (1/2)du=2t$^{3}$+t dt. I cannot find out how to relate that to the numerator (although I am so close).

2. Dec 1, 2011

### Staff: Mentor

Re: Evaluate integral of a polynomial over another polynomial (complicated substituti

Do you know partial fractions decomposition? That seems to me to be the way to go. Using that technique you rewrite (t3 + t - 1)/(t2(t2 + 1) as the sum of three rational expressions of the form
$$\frac{A}{t} + \frac{B}{t^2} + \frac{Ct + D}{t^2 + 1}$$

The idea is to find constants A, B, C, D so that the new representation is identically equal to the original rational expression. Once you find the constants, then integrate the sum of simpler functions.