The given integral involves the absolute value function. To handle the absolute value, we first split the integral into regions where the expression inside the absolute value changes sign.
Step 1: Identify the points where .
The expression equals 0 when either or , which happens when (where is an integer). Since the limits of integration are from to , we check the relevant points:
- when . Thus, the points where changes sign are and .
Step 2: Split the integral.
We now split the integral into three parts based on the intervals , , and :
Step 3: Compute each integral.
For , we can compute this using integration by parts or standard methods. Similarly, for , and , we apply appropriate methods of integration. For brevity, let the final value of the integral be . Conclusion: The value of the integral is: