Using the Maclaurin series for cos(x)=1−x22!+x44!−x66!+⋯\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdotscos(x)=1−2!x2+4!x4−6!x6+⋯, which series represents ∫0xcos(t) dt\displaystyle \int_0^x \cos(t)\,dt∫0xcos(t)dt?
x−x33⋅2!+x55⋅4!−⋯x - \frac{x^3}{3 \cdot 2!} + \frac{x^5}{5 \cdot 4!} - \cdotsx−3⋅2!x3+5⋅4!x5−⋯
x−x33!+x55!−x77!+⋯x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdotsx−3!x3+5!x5−7!x7+⋯
1−x24+x496−⋯1 - \frac{x^2}{4} + \frac{x^4}{96} - \cdots1−4x2+96x4−⋯
sin(x)\sin(x)sin(x) evaluated from 0 to xxx, which equals the antiderivative