Approximate \( \int \limits _{0}^{.1} \cos (x^4) dx \) with error magnitude less than \( 10^{-8} \)

\( \int \limits _{0}^{.1} \cos (x^4) dx = \int \limits _{0}^{.1} \left ( 1 - \frac{x^8}{2} + \frac{x^{16}}{24} + \cdots \right ) dx = \left [ x - \frac{x^9}{18} + \frac{x^{17}}{408}\cdots \right ] _0 ^{0.1} = \left ( .1 - \frac{.1^9}{18} + \cdots \right ) - \left ( 0 + 0 \cdots \right ) \approx .1 \)

With \( \left | \mbox{error} \right | < \frac{.1^9}{18}<10^{-8} \)