Write the integral as a sum of integrals without absolute values and evaluate. (Use decimal notation. Give your answer to three decimal places.) \[ \int_{\pi / 4}^{\pi}|\cos (x)| d x \approx \]

Step 1 ：The absolute value of a function can be thought of as the function itself when the function is positive, and the negative of the function when the function is negative. In the case of the cosine function, it is positive in the first and fourth quadrants, and negative in the second and third quadrants.

Step 2 ：Therefore, we can split the integral into two parts: one from \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \) where \( \cos(x) \) is positive, and one from \( \frac{\pi}{2} \) to \( \pi \) where \( \cos(x) \) is negative. We can then evaluate these two integrals separately.

Step 3 ：Calculate the first integral from \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \), the result is \( 1 - \frac{\sqrt{2}}{2} \).

Step 4 ：Calculate the second integral from \( \frac{\pi}{2} \) to \( \pi \), the result is 1.

Step 5 ：Add the two results together to get the final integral, which is \( 2 - \frac{\sqrt{2}}{2} \).

Step 6 ：\(\boxed{2 - \frac{\sqrt{2}}{2}}\) is the final answer.