S ( x ) = ∫ 0 x sin ⁡ ( t 2 ) d t = ∑ n = 0 ∞ ( − 1 ) n x 4 n + 3 ( 2 n + 1 ) ! ( 4 n + 3 ) C ( x ) = ∫ 0 x cos ⁡ ( t 2 ) d t = ∑ n = 0 ∞ ( − 1 ) n x 4 n + 1 ( 2 n ) ! ( 4 n + 1 ) {\displaystyle {\begin{aligned}S(x)&=\int _{0}^{x}\sin \left(t^{2}\right)\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+3}}{(2n+1)!(4n+3)}}\\C(x)&=\int _{0}^{x}\cos \left(t^{2}\right)\,dt=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{4n+1}}{(2n)!(4n+1)}}\end{aligned}}}