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    Hello, bit rusty on integration, but I have this integral: x^2/(x^2-2x+5) dx, and I've been told I have to use u-substitution. Tried doing u= x^2-2x+5 but haven't managed to get anywhere so far. Any tips?
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    (Original post by jordanwu)
    Hello, bit rusty on integration, but I have this integral: x^2/(x^2-2x+5) dx, and I've been told I have to use u-substitution. Tried doing u= x^2-2x+5 but haven't managed to get anywhere so far. Any tips?
    Complete the square on the denominator, THEN look for an appropriate substitution.
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    (Original post by jordanwu)
    Hello, bit rusty on integration, but I have this integral: x^2/(x^2-2x+5) dx, and I've been told I have to use u-substitution. Tried doing u= x^2-2x+5 but haven't managed to get anywhere so far. Any tips?
    Try doing long division (shouldn't be too difficult) first, then complete the square on the denominator as RDKGames suggested.
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    (Original post by RDKGames)
    Complete the square on the denominator, THEN look for an appropriate substitution.
    Ok so I have (u+1)^2 / u^2 + 4 du so far?
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    (Original post by jordanwu)
    Ok so I have (u+1)^2 / u^2 + 4 du so far?
    Expand the numerator, and say that u^2=(u^2+4)-4 to get (u^2+4)+2u-3 on the numerator. Then split your fraction into 3 separate ones with numerators u^2+4, 2u and -3 with each denominator being, yknow, unchanged. Then the first term simplifies to a constant, the second term is a matter of using the ln rule to integrate, and the third integrates to some trig function which you should recognise.
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    (Original post by RDKGames)
    Expand the numerator, and say that u^2=(u^2+4)-4 to get (u^2+4)+2u-3 on the numerator. Then split your fraction into 3 separate ones with numerators u^2+4, 2u and -3 with each denominator being, yknow, unchanged. Then the first term simplifies to a constant, the second term is a matter of using the ln rule to integrate, and the third integrates to some trig function which you should recognise.
    I ended up getting x-1 + ln((x-1)^2 + 4) - (3/2)tan^-1((x-1)/2) + C, not sure if I messed up somewhere or not
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    (Original post by jordanwu)
    I ended up getting x-1 + ln((x-1)^2 + 4) - (3/2)tan^-1((x-1)/2) + C, not sure if I messed up somewhere or not
    Looks good.
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    (Original post by jordanwu)
    I ended up getting x-1 + ln((x-1)2 + 4) - (3/2)tan-1((x-1)/2) + C, not sure if I messed up somewhere or not
    it looks correct :borat:
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    (Original post by RDKGames)
    Looks good.
    Ok, so the next one is this: 1/(5-2cosx) dx, and it says to use x= a tan^-1(u) for some appropriate constant a. Not sure where to start with this
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    (Original post by jordanwu)
    Ok, so the next one is this: 1/(5-2cosx) dx, and it says to use x= a tan^-1(u) for some appropriate constant a. Not sure where to start with this
    Why don't you use x=a arctan u and then choose an a halfway through the calculation when you see what is most useful?
 
 
 
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