Use a series to estimate the following integral's value with an error of magnitude less than $10^{-5}$ \[ \int^{0.6} \sin x^{2} d x \]

Step 1 ：The integral of \(\sin(x^2)\) does not have a standard form. However, we can use the power series expansion of the sine function to approximate the integral.

Step 2 ：The power series expansion of \(\sin(x)\) is given by: \[\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]

Step 3 ：We can substitute \(x^2\) into this series to get the series expansion of \(\sin(x^2)\): \[\sin(x^2) = x^2 - \frac{(x^2)^3}{3!} + \frac{(x^2)^5}{5!} - \frac{(x^2)^7}{7!} + \cdots\]

Step 4 ：We can then integrate this series term by term from 0 to 0.6 to approximate the value of the integral. We can keep adding terms until the magnitude of the last term added is less than \(10^{-5}\).

Step 5 ：The approximate value of the integral \(\int^{0.6} \sin x^{2} d x\) is \(\boxed{0.3387888}\).