Express the indefinite integral $\int\sin x^2~dx$ as a power series

Question

What does this mean? I never saw this in my class/notes so I don't understand the conversion from integral to power series.

Also if the integral were defined from $0$ to $1$, what new steps do I add?

Answer 1

$\int\sin x^2~dx$

$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n(x^2)^{2n+1}}{(2n+1)!}dx$

$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{4n+2}}{(2n+1)!}dx$

$=\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{4n+3}}{(2n+1)!(4n+3)}+C$

$\therefore\int_0^1\sin x^2~dx$

$=\left[\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{4n+3}}{(2n+1)!(4n+3)}\right]_0^1$

$=\sum\limits_{n=0}^\infty\dfrac{(-1)^n}{(2n+1)!(4n+3)}$