3.2 Use Maclaurin series expansion to approximate the value of the following integral
1
𝐼 = ∫ 𝑒𝑥 cos 𝑥𝑑𝑥.
0
The series is 1 + x - x^3/3 - x^4/6 - x^5/30 + ...
so, integrating each term, we get
x + x^2/2 - x^4/12 - x^5/30 - x^6/180 + ...
summing that from 0 to 1, we have
(1 + 1/2 - 1/12 - 1/30 - 1/180 + ...) - (1+0+0+...)
= -1/2 - 1/12 - 1/30 - 1/180 + ...
Use as many terms as you like to get the desired accuracy.