# Power Series for Indefinite Integral

• April 25th 2010, 08:42 PM
leilani13
Power Series for Indefinite Integral
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.
• April 25th 2010, 08:53 PM
Prove It
Quote:

Originally Posted by leilani13
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.

$\cos{X} = \sum_{n = 0}^{\infty}\left[\frac{(-1)^nX^{2n}}{(2n)!}\right]$.

So $\cos{(x^2)} = \sum_{n = 0}^{\infty}\left[\frac{(-1)^n(x^2)^{2n}}{(2n)!}\right]$

$= \sum_{n = 0}^{\infty}\left[\frac{(-1)^nx^{4n}}{(2n)!}\right]$.

So now find

$\int{\cos{(x^2)}\,dx} = \int{\sum_{n = 0}^{\infty}\left[\frac{(-1)^nx^{4n}}{(2n)!}\right]\,dx}$.