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Thread: Integration help

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    Integration help

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

    Thanks for any help
    2016 HSC (Accelerated)
    // 2 Unit Maths // Studies of Religion 1 //

    2017 HSC
    // Biology // Physics // Maths Extension 1 // Maths Extension 2 // English Advanced //

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    Re: Integration help

    Quote Originally Posted by pikachu975 View Post
    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).
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    Re: Integration help

    Quote Originally Posted by InteGrand View Post
    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?
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    // 2 Unit Maths // Studies of Religion 1 //

    2017 HSC
    // Biology // Physics // Maths Extension 1 // Maths Extension 2 // English Advanced //

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    Re: Integration help

    Quote Originally Posted by pikachu975 View Post
    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?
    pikachu975 likes this.

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