Chapter 10: Infinite Sequences and Series - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 25

$| Error| \lt 0.000013$

Integrate the integral with respect to $ x $ as follows: $ f(x)=\int_0^{x} \sin t^2 dt \\=\int_0^{x} [t^2-\dfrac{t^6}{3 !}+\dfrac{t^{10}}{5!}-...] \space dt \\=\dfrac{x^3}{3}-\dfrac{x^{7}}{7 \cdot 3 !}+\dfrac{x^{11}}{11 \cdot 5!}- ....$ Now, $| Error| \lt \dfrac{1}{15\cdot 7 !} \approx 0.000013$ or, $| Error| \lt 0.000013$