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Math Help - How do you spot it's a substitution

  1. #1
    Junior Member
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    How do you spot it's a substitution

    \int^{9}_{5} \frac{x}{\sqrt{(x-5)^{2}+4^{2}}}

    Now, according to my mark scheme this is not by parts, but using the substitution u = x-5. How on earth was I ment to spot that?? How would you maths gurus spot something like that

    Thanks
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  2. #2
    MHF Contributor Amer's Avatar
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    Jordan
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    Quote Originally Posted by thomas49th View Post
    \int^{9}_{5} \frac{x}{\sqrt{(x-5)^{2}+4^{2}}}

    Now, according to my mark scheme this is not by parts, but using the substitution u = x-5. How on earth was I ment to spot that?? How would you maths gurus spot something like that

    Thanks
    \int_{5}^{9} \frac{x}{\sqrt{(x-5)^{2}+4^{2}}}

    u=x-5 du=dx

    \int_{0}^{4} \frac{u+5}{u^2+4^{2} } du

    let 2tan(y) = u \Rightarrow 2sec^2y dy =du

    2tan(y)=u \Rightarrow 4tan^2 y =u^2 \Rightarrow 4(sec^2y -1 ) =u^2 \Rightarrow sec^2y = u^2+4

    you will have

    \int \left(\frac{2tan(y) +5 }{sec^2 y }\right)\left(2sec^2y dy \right) dy

    can you continue
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