How Do You Solve This Complex Integral?

[skyturnred]

Homework Statement

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

Homework Equations

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).

Can someone please help? Thanks!

 

[Mark44]

Homework Statement

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

Homework Equations

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).

Can someone please help? Thanks!

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.