Evaluate : `int_(0)^(pi)(|cos x| + |sin x|)dx`.

Answer

To evaluate the integral \( I = \int_{0}^{\pi} (|\cos x| + |\sin x|) \, dx \), we can break it down into two parts: the integral of \( |\cos x| \) and the integral of \( |\sin x| \).

### Step 1: Split the Integral
We can write the integral as:
\[ I = \int_{0}^{\pi} |\cos x| \, dx + \int_{0}^{\pi} |\sin x| \, dx \]

**Hint:** Remember that the absolute value function can change the sign of the function based on the intervals where the function is positive or negative.

### Step 2: Analyze \( |\sin x| \)
For \( \sin x \) in the interval \( [0, \pi] \):
- \( \sin x \) is non-negative (positive) for \( x \in [0, \pi] \).

Thus, we can write:
\[ \int_{0}^{\pi} |\sin x| \, dx = \int_{0}^{\pi} \sin x \, dx \]

**Hint:** Since \( \sin x \) is positive in the entire interval from \( 0 \) to \( \pi \), we can drop the absolute value.

### Step 3: Analyze \( |\cos x| \)
For \( \cos x \) in the interval \( [0, \pi] \):
- \( \cos x \) is positive for \( x \in [0, \frac{\pi}{2}] \) and negative for \( x \in [\frac{\pi}{2}, \pi] \).

Thus, we can split the integral:
\[ \int_{0}^{\pi} |\cos x| \, dx = \int_{0}^{\frac{\pi}{2}} \cos x \, dx + \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx \]

**Hint:** The negative sign for \( \cos x \) in the second integral accounts for the fact that \( \cos x \) is negative in that interval.

### Step 4: Evaluate the Integrals
Now we evaluate each integral separately.

1. **For \( \int_{0}^{\frac{\pi}{2}} \cos x \, dx \)**:
\[ \int \cos x \, dx = \sin x \quad \text{so} \quad \int_{0}^{\frac{\pi}{2}} \cos x \, dx = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

2. **For \( \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx \)**:
\[ \int -\cos x \, dx = -\sin x \quad \text{so} \quad \int_{\frac{\pi}{2}}^{\pi} -\cos x \, dx = -\sin(\pi) - (-\sin\left(\frac{\pi}{2}\right)) = 0 - (-1) = 1 \]

3. **For \( \int_{0}^{\pi} \sin x \, dx \)**:
\[ \int \sin x \, dx = -\cos x \quad \text{so} \quad \int_{0}^{\pi} \sin x \, dx = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2 \]

### Step 5: Combine the Results
Now, we can combine the results:
\[ I = \int_{0}^{\pi} |\cos x| \, dx + \int_{0}^{\pi} |\sin x| \, dx = (1 + 1) + 2 = 4 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{4} \]