3) Assume that equals its Maclaurin series for all x.
Use the Maclaurin series for to evaluate the integral . Your answer will be an infinite series. Use the first two terms to estimate its value.
I am not sure how to do this!
I'm sorry about that, but I cannot find a serious mistake in my calculation. If I ask a CAS to integrate numerically I get the answer .
Now this does not fit exactly what I have found, namely , but that's not particularly surprising because the next term in the series gives .
If that's the case, then I take it that your teacher must be mistaken, or, another possibility, not telling you the truth. Why could that be? - Well, I remember that during my studies I once was told that my solution of a somewhat more difficult exercise was wrong. I immediately startet arguing with the teacher. After a few exchanges as to why I thought his objections were not valid, he began to smile and told me that he now really did believe that I was right.
So at that time, my teacher was simply testing whether I had found the solution to this exercise on my own (or at least understood the solution so well that I was able to argue for it convincingly). If a student gets the solution from someone else however, he or she typically will not immediately try to argue for its correctness, but instead rather quickly withdraw...