# Evaluating an indefinite integral as an infinite series

• Mar 15th 2013, 10:46 AM
RobertXIV
Evaluating an indefinite integral as an infinite series
Greetings! I'm not sure how to approach this question, it's asking that I rewrite this:

(sin(x)-x)/x^3 dx

as an infinite series. Any help is appreciated!
• Mar 15th 2013, 12:15 PM
HallsofIvy
Re: Evaluating an indefinite integral as an infinite series
Write sin(x) as a Taylor's series. Then you can subtract x, divide by $x^3$ and integrate term by term.
• Mar 15th 2013, 03:32 PM
RobertXIV
Re: Evaluating an indefinite integral as an infinite series
Ok let me see...
sinx as a Taylor series would be the sum of ((-1)^n/(2n+1)!)*x^(2n+1)

so then would we have (((-1)^n/(2n+1)!)*x^(2n+1)/x^3) - (x/x^3)?
• Mar 15th 2013, 04:03 PM
Prove It
Re: Evaluating an indefinite integral as an infinite series
Yes, and surely you can now simplify...
• Mar 15th 2013, 05:30 PM
HallsofIvy
Re: Evaluating an indefinite integral as an infinite series
It might make more sense if you write it out as $sin(x)= x- x^3/3!+ x^5/5!- x^7/7!+ x^9/9!- \cdot\cdot\cdot$.
Then $sin(x)- x= - x^3/3!+ x^5/5!- x^7/7!+ x^9/9!- \cdot\cdot\cdot$
and $\frac{sin(x)- x}{x^3}= -1+ x^2/5!- x^4/7!+ x^6/9!- \cdot\cdot\cdot$
Integrate that.
• Mar 16th 2013, 11:37 AM
RobertXIV
Re: Evaluating an indefinite integral as an infinite series
Ah! Alright that makes much more sense now. Thank you very much!