NO, you are NOT supposed to do that!
Read the problem again- it said "use the MacLaurin series of cos(x)" and you did not do that. The MacLaurin series of cos(x) is . The first four non-zero terms of that are .
Replaced cos(x) in and you will see why I gave the first four terms of cos(x) rather than just the first three.
That is just a 6th degree polynomial. Integrate the polynomial between 0 and 1.