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    I'm back with another integral, it's been a while, posted this somewhere else but may as well make a thread.

    (i) Integrate with respect to  x  \displaystyle \frac{1}{x(1-x^7)} .

    (ii) Hence, or otherwise, evaluate the integral
     \displaystyle \int _{1/2}^{\pi /4} \frac{1}{x(1-x^7)} dx ,
    leaving your answer in a simplified exact form.
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    (Original post by Ano123)
    I'm back with another integral, it's been a while, posted this somewhere else but may as well make a thread.

    (i) Integrate with respect to  x  \displaystyle \frac{1}{x(1-x^7)} .
    Note that:

    \displaystyle

\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \ln \left(1 - \frac{1}{x^7}\right) = -\frac{7}{x(1-x^7)}\end{equation*}
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    (Original post by Zacken)
    Note that:

    \displaystyle

\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \ln \left(1 - \frac{1}{x^7}\right) = -\frac{7}{x(1-x^7)}\end{equation*}
    How did you do it by inspection? Seen similar types before or just recognised it was the derivative of a relatively simple function?
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    (Original post by Ano123)
    How did you do it by inspection? Seen similar types before or just recognised it was the derivative of a relatively simple function?
    Just recognised. Shouldn't be too hard to do as a geometric series with a first term \frac{1}{x} and common ratio x^7 either.
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    (Original post by Zacken)
    Just recognised. Shouldn't be too hard to do as a geometric series with a first term \frac{1}{x} and common ratio x^7 either.
    I should have made it more difficult with a square root or something.
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    (Original post by Ano123)
    I should have made it more difficult with a square root or something.
    *shrugs* indefinite integrals are boring. It means that an elementary anti-derivative exists which is a buzzkill assumption.
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    (Original post by Zacken)
    *shrugs* indefinite integrals are boring. It means that an elementary anti-derivative exists which is a buzzkill assumption.
    What would you prefer then?
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    (Original post by Ano123)
    What would you prefer then?
    Definite ones, then elementary antiderivatives need not exist.
 
 
 
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