1. ## Integration help

Integrate (x-1)(x+1)dx / (x-2)(x-3)

Thanks for any help

2. ## Re: Integration help

Originally Posted by pikachu975
Integrate (x-1)(x+1)dx / (x-2)(x-3)

Thanks for any help
It's a quadratic over a quadratic, so use long division first and then the "remainder part" may be integrated using partial fraction decomposition (the quotient will just be a polynomial, which is easy to integrate).

3. ## Re: Integration help

Originally Posted by InteGrand
It's a quadratic over a quadratic, so use long division first and then the "remainder part" may be integrated using partial fraction decomposition (the quotient will just be a polynomial, which is easy to integrate).
I accidentally split the numerator before I did partial fractions so I had:

x + integral (5x-7)dx/(x^2 - 5x+6)

So I split it up into:

x + integral (5/2 * (2x-5) + 11/2)/(x^2 - 5x + 6)

x + 5/2 * integral (2x-5)/(x^2 - 5x+6) + 11/2 * integral (dx/(x-3)(x-2))

x + 5/2 * ln|x^2 - 5x + 6| + 11/2 * integral (dx/(x-3)(x-2)) and then I did partial fractions on this

And got A = 1, B = -1

=> x + 5/2 * ln|x^2 - 5x + 6| + 11/2 * integral (1/(x-3) - 1/(x-2) dx
= x + 5/2 * ln|x^2 - 5x + 6| + 11/2 * ln|(x-3)/(x-2)| + C

But the answer is x - 3ln(x-2) +8ln(x-3) + C

Did I do a mistake in the working?

4. ## Re: Integration help

Originally Posted by pikachu975
I accidentally split the numerator before I did partial fractions so I had:

x + integral (5x-7)dx/(x^2 - 5x+6)

So I split it up into:

x + integral (5/2 * (2x-5) + 11/2)/(x^2 - 5x + 6)

x + 5/2 * integral (2x-5)/(x^2 - 5x+6) + 11/2 * integral (dx/(x-3)(x-2))

x + 5/2 * ln|x^2 - 5x + 6| + 11/2 * integral (dx/(x-3)(x-2)) and then I did partial fractions on this

And got A = 1, B = -1

=> x + 5/2 * ln|x^2 - 5x + 6| + 11/2 * integral (1/(x-3) - 1/(x-2) dx
= x + 5/2 * ln|x^2 - 5x + 6| + 11/2 * ln|(x-3)/(x-2)| + C

But the answer is x - 3ln(x-2) +8ln(x-3) + C

Did I do a mistake in the working?
$\noindent Rewrite the quadratic in a logarithm in your answer as (x-2)(x-3), then use log laws to split both logarithms and it should simplify down to the right answer.$

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