Chapter 6.4, Problem 216E

In the following exercises, find the Maclaurin series of $F(x)={\displaystyle \underset{0}{\overset{x}{\int }}f(t)dt}$by integrating the Maclaurin series of f term b term. If f is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.

216. $F(x)={\displaystyle \underset{0}{\overset{x}{\int }}\frac{1-\cos t}{t^2}dt;}$ $f(t)=\frac{1-\cos t}{t^2}={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{(-1)}^n\frac{t^2{}^n}{(2n+2)!}}$