## unimatix one year ago Integral (3x+1)/(x^2-3x+2) dx

1. unimatix

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2. anonymous

did you try partial fractions?

3. unimatix

No hold I'll try that.

4. unimatix

Not sure how to do that in this case.

5. unimatix

Maybe like a 1/3 x?

6. anonymous

factor the denominator to $$(x-1)(x-2)$$ Then the original fraction can be written as $\frac{ A }{ x-1 }+\frac{ B }{ x-2 }$

7. anonymous

$\frac{ 3x+1 }{ (x-1)(x-2) }=\frac{ A }{ x-1 }+\frac{ B }{ x-2 }$

8. unimatix

Okay I got that.

9. unimatix

I'm watching a video on partial fractions because it's been so long since I've seen them.

10. Jhannybean

$\frac{3x+1}{x^2-3x+2} = \frac{3x+1}{(x-2)(x-1)}$$\frac{3x+1}{(x-2)(x-1)} = \frac{A}{(x-2)}+\frac{B}{(x-1)}$$3x+1=A(x-1)+B(x-2)$Yu could use a system of equations or the "zero" method to solve for the x's

11. anonymous

add on the right then set thh denominators equal. $A(x-1)+B(x-2)=3x+1$ Equate the coefficients and solve the system $A+B=3$ $-A-2B=1$

12. anonymous

And then once you have A and B the integral becomes $\int\limits \left( \frac{ A }{ x-1 }+\frac{ B }{ x-2 } \right)dx$

13. Jhannybean

Let's say we took $$x=1$$, then $3(1)+1=A(1-1)+B(1-2)$$4=0-B$$B=-4$$3x+1=A(x-1)+B(x-2)$$3(2)+1=A(2-1)+0$$7=A$

14. amistre64

not sure if partials are needed, what is the derivative of the bottom to start with? can we modify the top?

15. Jhannybean

Basically trying to find what makes A and B = 0... etc.

16. Jhannybean

$\int \left(\frac{7}{x-1}-\frac{4}{x-2}\right)dx$... Trying to refresh my memory on Partial Fraction Decomposition as well. I found PFD easier because the polynomial at the bottom could be factored, and factoring leads me to partial fractions.. etc.

17. amistre64

2x-3: 3x+1+(-x+4+x-4) 2x-3: 2x-3+(x+4) meh, would still need it for part of it i spose ...

18. Jhannybean

$7\int \left(\frac{1}{x-1}\right)dx -4\int \left(\frac{1}{x-2}\right)dx$ Im guessing?

19. Jhannybean

@unimatix Are you capable of continuing from here?

20. unimatix

I think so! Thanks for all your help everyone!

21. Jhannybean

And can you show me how you would integrate this? Just want to make sure you do it right.