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Math Help - Two (Hopefully the last two!) Integration Questions

  1. #1
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    Two (Hopefully the last two!) Integration Questions

    These questions share a similar theme.

    1) Integral of (x-1)dx/(x^2 - 4x + 5)

    I have tried dividing this problem into two fractions, without result. I thought maybe difference of squares somehow on the bottom, x^2 - 4x + 4 + 1, but what then?

    2) xdx/x^4 + x^2 + 1

    I set u = x^2
    du = 2x
    = 1/2 Integral of du/u^2+u+1

    And I STILL can't solve it! It is in a really simple form now, 1/u^2 + u + 1, what am I doing wrong?

    Seems like it would be obvious, but it's evading me.

    Sigh. Why is this not obvious to me? V. frustrating
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  2. #2
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    \int\frac{x-1}{x^2-4x+5}dx


    Well you know that:

    x^2-4x+5=x^2-4x+4 +1 = (x-2)^2+1

    so:

    \int\frac{x-1}{(x-2)^2+1}

    so let's let u = x-2

    so du= dx and u+1=x-1

    \int\frac{u+1}{u^2+1}

    Then I would do a second substitution with trigonometry:

    let u = tan\theta

    du = \frac{1}{cos^2\theta}d\theta

    \int\frac{tan\theta+1}{tan^2\theta +1}*\frac{1}{cos^2\theta}d\theta

    \int\frac{tan\theta+1}{\frac{1}{cos^2\theta}}*\fra  c{1}{cos^2\theta}d\theta

    \int (tan\theta+1)d\theta
    Last edited by seld; September 15th 2009 at 09:23 PM. Reason: bad latex
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  3. #3
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    Ah, it was the difference of squares trick. I don't feel confident using it, it always seems suspicious to me!

    Thanks, though, great advice and v. helpful!
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  4. #4
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    Well the truth is that I went through calculus I, II and III with only power rule, product rule, derivative of sine and cosine memorized.

    Something useful to remember with trig substitutions though:

    sin^2\theta+cos^2\theta=1

    divide by cos^2 on both sides:
    \frac{sin^2\theta}{cos^2\theta}+\frac{cos^2\theta}  {cos^2\theta}=\frac{1}{cos^2\theta}

    tan^2\theta+1=sec^2\theta



    sin^2\theta+cos^2\theta=1

    divide by sin^2 on both sides:
    \frac{sin^2\theta}{sin^2\theta}+\frac{cos^2\theta}  {sin^2\theta}=\frac{1}{sin^2\theta}

    1+cot^2\theta=csc^2\theta


    And you don't have to memorize is it tan^2 -1 = csc^2 ? or whatever just straight and simple.
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  5. #5
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    \frac{u+1}{u^{2}+1}=\frac{u}{u^{2}+1}+\frac{1}{u^{  2}+1}.
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