Thread: Integral as a power series

1. Integral as a power series

Hey everyone,

I do not know how to express this integral as a power series:

$\int{\frac{x-arctan(x)}{x^3}$

I cannot figure out how to get this into the right form. I am a bit confused. Any help is appreciated.
Thanks

2. Originally Posted by evant8950
Hey everyone,

I do not know how to express this integral as a power series:

$\int{\frac{x-arctan(x)}{x^3}$

I cannot figure out how to get this into the right form. I am a bit confused. Any help is appreciated.
Thanks
$\displaystyle \arctan{x} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...$

$\displaystyle x - \arctan{x} = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ...
$

$\displaystyle \frac{x - \arctan{x}}{x^3} = \frac{1}{3} - \frac{x^2}{5} + \frac{x^4}{7} - \frac{x^6}{9} + ...
$

integrate the last series.

3. Originally Posted by skeeter
$\displaystyle \arctan{x} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ...$

$\displaystyle x - \arctan{x} = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ...
$

$\displaystyle \frac{x - \arctan{x}}{x^3} = \frac{1}{3} - \frac{x^2}{5} + \frac{x^4}{7} - \frac{x^6}{9} + ...
$

integrate the last series.

I understand that the $\int{\frac {1}{1+x^2}$ is equal to arctanx. Then I converted that to a power series and integrated $\sum_{n=0}^\infty\((-1)^n\frac {x^{2n +1}}{2n+1}}+C$

Where do I go from here?

4. Read again the post above...

1. Replace the arctan(x) function in your integral with power series of arctan(x)

2. Understand what skeeter wrote.

3. Integrate term-by-term (Why you allowed to do this?)

5. I understand #2 on what skeeter wrote. I understand on #3 when the $x^3$ is multiplied back. What happened to the x in #2? I see the negative got distributed, but why is x not being added to each term?

Thanks

6. Originally Posted by evant8950
I understand #2 on what skeeter wrote. I understand on #3 when the $x^3$ is multiplied back. What happened to the x in #2? I see the negative got distributed, but why is x not being added to each term?

Thanks
why would x be added to each term? ... just subtract the series for arctan(x) from x.

$\displaystyle x - \arctan{x} = x - \left(x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + ... \right) = x - x + \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ... \, = \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} - \frac{x^9}{9} + ...$

7. Thanks! I got it now.