1. Can you compute ?
Well, first we consider the antiderivative for …which doesn’t exist.
What do we do? Cry. No, what I mean is, use Taylor series!
2. We should recall the Taylor series for cosine is
So we plug in to the series and we get
Now we plug (2) into our integral, we get
This is a more manageable calculation, but we end up with an infinite series
How do we evaluate this?
3. We usually truncate it at some point. For example, we know each term decreases (the series converges absolutely). So if we stop at the fifth term, we are throwing away terms smaller than
So to about 7 digits, this gives us a good approximation
where we truncate to 7 digits (i.e., we throw away everything after 7 digits, since it’d be corrected by higher order terms in the series).