Definite Integral

We need to evaluate the integral . The function is piecewise defined, so we split the integral into two parts: For , , and for , . 1. Evaluate the first integral, where for : 2. Evaluate the second integral, where for : Now, sum the results:

The correct option is (C) :

Step 1: Break the integral into two parts The function behaves differently for positive and negative values:
- For , , so .
- For , , so .
Thus, we split the integral at : Step 2: Evaluate the integrals Evaluating the first integral: Evaluating the second integral: Step 3: Compute the final result Summing both integrals: Thus, the integral evaluates to , confirming option (E).