Question Statement:
A function $f(x)$ is defined by $f(x) = \frac{1}{2}(x + |x|)$. The value of $\int_{-1}^{1} f(x)\,dx$ is _________ (round off to 1 decimal place).
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2019 Q.24 / AE Q.14)**
**1. Intuitive Physical Setup & Given Parameters**:
- Function: $f(x) = \frac{1}{2}(x + |x|)$
- Integral limits: from $x = -1$ to $x = 1$.
**2. Step-by-Step Mathematical Derivation**:
1. **Piecewise Definition of $|x|$**:
- For $x < 0$: $|x| = -x \implies f(x) = \frac{1}{2}(x - x) = 0$
- For $x \ge 0$: $|x| = x \implies f(x) = \frac{1}{2}(x + x) = x$
(This is the standard Ramp function $R(x) = \max(0, x)$).
2. **Splitting the Integral at $x = 0$**:
$\int_{-1}^{1} f(x)\,dx = \int_{-1}^{0} 0\,dx + \int_{0}^{1} x\,dx$
3. **Evaluating the Non-Zero Integral**:
$\int_{0}^{1} x\,dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1^2}{2} - 0 = \frac{1}{2} = 0.5$
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Exact Analytical Evaluation**: **0.5** (Acceptable Range: [0.5, 0.5]).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **0.5**
- **Exam Rule**: Whenever $|x|$ appears in a definite integral, split the domain across $x = 0$.
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