Find a power series for the indefinite integral of cos(x^2) dx. Use both sigma notation and expanded form, including the first four nonzero terms of the series.
$\displaystyle \cos{X} = \sum_{n = 0}^{\infty}\left[\frac{(-1)^nX^{2n}}{(2n)!}\right]$.
So $\displaystyle \cos{(x^2)} = \sum_{n = 0}^{\infty}\left[\frac{(-1)^n(x^2)^{2n}}{(2n)!}\right]$
$\displaystyle = \sum_{n = 0}^{\infty}\left[\frac{(-1)^nx^{4n}}{(2n)!}\right]$.
So now find
$\displaystyle \int{\cos{(x^2)}\,dx} = \int{\sum_{n = 0}^{\infty}\left[\frac{(-1)^nx^{4n}}{(2n)!}\right]\,dx}$.