# Math Help - Evaluating an indefinite integral as an infinite series

1. ## 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!

2. ## 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.

3. ## 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)?

4. ## Re: Evaluating an indefinite integral as an infinite series

Yes, and surely you can now simplify...

5. ## 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.

6. ## Re: Evaluating an indefinite integral as an infinite series

Ah! Alright that makes much more sense now. Thank you very much!