Partial Fractions: Integrating a Problem

[schapman22]

Homework Statement

We were given a worksheet to integrate some problems using partial fractions. This one however I cannot figure out what to do with it. This is the problem.

∫(x³-4x²+x+6)/(x²-x+2) dx

The Attempt at a Solution

using long division i got ∫(x-3) (-4(x-3)/(x²-x+2) dx
split it up into ∫xdx - 3∫dx - 4∫(x-3)/(x²-x+2)
now I don't know what to do with the x²-x+2?
any suggestions on how to proceed from here? Thanks in advance!

 

[SammyS]

Homework Statement

We were given a worksheet to integrate some problems using partial fractions. This one however I cannot figure out what to do with it. This is the problem.

∫(x³-4x²+x+6)/(x²-x+2) dx

The Attempt at a Solution

using long division i got ∫(x-3) + (-4(x-3)/(x²-x+2) dx
split it up into ∫xdx - 3∫dx - 4∫(x-3)/(x²-x+2)
now I don't know what to do with the x²-x+2?
any suggestions on how to proceed from here? Thanks in advance!

What you have done so far looks correct.

For the integral \displaystyle \int 2\frac{2x-6}{x^2-x+2}\,dx\,, notice that the derivative of the denominator is 2x-1.

Split the integrand up into \displaystyle <br /> 2\frac{2x-1}{x^2-x+2}+2\frac{-5}{x^2-x+2}\,.

So you get two integrals. The first results in a logarithm. For the second one, complete the square in the denominator.

 

[schapman22]

So I'm allowed to multiple the integrand by 2?

 

[schapman22]

thank you