Integral Homework Help: Solving (x+2)/(x^2-2x+3) dx | Step-by-Step Guide

[grimireaper]

Homework Statement

Integrate

Homework Equations

I started with (2x^2+3)/(x^3-2x^2+3x) dx

The Attempt at a Solution

Got dx/x + (x+2)/(x^2-2x+3) dx using the A/x + (Bx+C)/(x^2-2x+3) method.

But I can't seem to solve (x+2)/(x^2-2x+3) dx

Possible solutions are welcome.

 

[LCKurtz]

I started with (2x^2+3)/(x^3-2x^2+3x) dx

and got dx/x + (x+2)/(x^2-2x+3) dx using the A/x + (Bx+C)/(x^2-2x+3) method.

But I can't seem to solve (x+2)/(x^2-2x+3) dx

Possible solutions are welcome.

Welcome to PF. You won't get solutions here, but you may get hints. Try this:$$
\frac {x+2}{x^2-2x+3} = \frac{(x-1)+3}{(x-1)^2+2}$$and see if that gives you any ideas.

 

[grimireaper]

I thought of that but the +2 at the bottom ruins it, because splitting up x-1 and 3 into 2 integrals won't help, as I can't cancel anything out

 

[LCKurtz]

I thought of that but the +2 at the bottom ruins it, because splitting up x-1 and 3 into 2 integrals won't help, as I can't cancel anything out

But you can break it up into two fractions and use different methods on them.

 

[lurflurf]

$$2x^2+3=(x^2-x)+(3x)+(x^2-2x+3)=\frac{x}{2}\left(\frac{x^3-2x^2+3x}{x}\right)^\prime+3\frac{x^3-2x^2+3x}{x^2-2x+3}+\frac{x^3-2x^2+3x}{x}$$

 

[grimireaper]

Got it solved. Thank you for the help.