The Student Room Group

Integral help??????!!!!????

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 x 1x(1x7) \displaystyle \frac{1}{x(1-x^7)} .

(ii) Hence, or otherwise, evaluate the integral
1/2π/41x(1x7)dx \displaystyle \int _{1/2}^{\pi /4} \frac{1}{x(1-x^7)} dx ,
leaving your answer in a simplified exact form.
Reply 1
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 x 1x(1x7) \displaystyle \frac{1}{x(1-x^7)} .


Note that:

Unparseable latex formula:

\displaystyle[br]\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \ln \left(1 - \frac{1}{x^7}\right) = -\frac{7}{x(1-x^7)}\end{equation*}

Reply 2
Original post by Zacken
Note that:

Unparseable latex formula:

\displaystyle[br]\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?
Reply 3
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 1x\frac{1}{x} and common ratio x7x^7 either.
Reply 4
Original post by Zacken
Just recognised. Shouldn't be too hard to do as a geometric series with a first term 1x\frac{1}{x} and common ratio x7x^7 either.


I should have made it more difficult with a square root or something.
Reply 5
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.
Reply 6
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?
Reply 7
Original post by Ano123
What would you prefer then?


Definite ones, then elementary antiderivatives need not exist.

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