Approximate the integral int (1 - cos x)/x dx using Taylor expansion

[thenewbosco]

I am supposed to find an approximation of this integral evaluated between the limits 0 and 1 using a taylor expansion for cos x:

$$\int \frac{1 - cos x}{x}dx$$

and given

$$cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}...$$

i should get a simple series similar to this for $$\frac{1 - cos x}{x}$$ and be able to simply integrate each term of the series and evaluate the integral for an approx. how do i find this series?

 

[Tzar]

Just sub in the series into the integral instead of cos (x). The 1's will cancel and the x at the bottom will decrease the power of each x on top by 1. Then integrate.