\int_0^x\frac{\sin t}{t}dt=\int_0^x\sum_{n=0}^{\infty}\frac{t^{2n}}{(2n+1)!}dt=\sum_{n=0}^{\infty}\int_0^x\frac{t^{2n}}{(2n+1)!}dt=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)(2n+1)!}