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Integration using series expansion

  1. Jul 27, 2009 #1
    Can't we integrate tanx/x dx using the series expansion of tan x?
     
  2. jcsd
  3. Jul 27, 2009 #2

    Cyosis

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    Homework Helper

    You can as long as the series converges between the limits of integration.

    [tex]
    \begin{align*}
    \int \frac{\tan x}{x} dx &=\int \frac{1}{x}\sum_{n=1}^\infty \frac{B_{2n} (-4)^n(1-4^n)}{(2n)!} x^{2n-1} dx \\
    &=\sum_{n=1}^\infty \left( \frac{B_{2n} (-4)^n(1-4^n)}{(2n)!} \int x^{2n-2} dx \right)
    \\
    &=\sum_{n=1}^\infty \frac{B_{2n} (-4)^n(1-4^n)}{(2n)!(2n-1)} x^{2n-1}
    \\
    &=x+\frac{x^3}{9}+\frac{2x^5}{75}+...\;\;\; \text{for} \;|x|<\frac{\pi}{2}
    \end{align}
    [/tex]
     
    Last edited: Jul 27, 2009
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