To evaluate the integral \( \int_0^{\pi} \min\{|\tan x|, |(4/\pi)x - 2|\} \, dx \), we need to analyze the two functions involved: \( |\tan x| \) and \( |(4/\pi)x - 2| \).
### Step 1: Analyze the Functions
1. **Function \( |\tan x| \)**:
- The function \( \tan x \) is positive in the interval \( (0, \frac{\pi}{2}) \) and becomes undefined at \( x = \frac{\pi}{2} \). Therefore, \( |\tan x| = \tan x \) for \( x \in (0, \frac{\pi}{2}) \) and \( |\tan x| \) is negative for \( x \in (\frac{\pi}{2}, \pi) \) but we consider the absolute value.
- As \( x \) approaches \( \frac{\pi}{2} \), \( |\tan x| \) approaches infinity.
2. **Function \( |(4/\pi)x - 2| \)**:
- This is a linear function that crosses the x-axis at \( x = \frac{\pi}{2} \).
- For \( x < \frac{\pi}{2} \), \( |(4/\pi)x - 2| = 2 - (4/\pi)x \).
- For \( x > \frac{\pi}{2} \), \( |(4/\pi)x - 2| = (4/\pi)x - 2 \).
### Step 2: Find Intersection Points
To find where \( |\tan x| = |(4/\pi)x - 2| \), we need to solve:
1. \( \tan x = 2 - \frac{4}{\pi}x \) for \( x \in [0, \frac{\pi}{2}) \)
2. \( \tan x = \frac{4}{\pi}x - 2 \) for \( x \in (\frac{\pi}{2}, \pi] \)
### Step 3: Evaluate the Integral
We will split the integral into three parts based on the intersection points found above, which we denote as \( a \) and \( b \):
1. From \( 0 \) to \( a \): \( \int_0^a \tan x \, dx \)
2. From \( a \) to \( b \): \( \int_a^b \left(2 - \frac{4}{\pi}x\right) \, dx \)
3. From \( b \) to \( \pi \): \( \int_b^{\pi} \tan x \, dx \)
### Step 4: Calculate Each Integral
1. **Integral from \( 0 \) to \( a \)**:
\[
\int_0^a \tan x \, dx = -\ln|\cos x| \bigg|_0^a = -\ln|\cos a| + \ln(1) = -\ln|\cos a|
\]
2. **Integral from \( a \) to \( b \)**:
\[
\int_a^b \left(2 - \frac{4}{\pi}x\right) \, dx = \left[2x - \frac{4}{\pi} \frac{x^2}{2}\right]_a^b = \left(2b - \frac{2}{\pi}b^2\right) - \left(2a - \frac{2}{\pi}a^2\right)
\]
3. **Integral from \( b \) to \( \pi \)**:
\[
\int_b^{\pi} \tan x \, dx = -\ln|\cos x| \bigg|_b^{\pi} = -\ln(0) + \ln|\cos b| = \infty - \ln|\cos b| \text{ (noting that } \cos(\pi) = -1\text{)}
\]
### Step 5: Combine Results
Combine the results of the three integrals to get the final answer.
### Final Answer
The final value of the integral \( \int_0^{\pi} \min\{|\tan x|, |(4/\pi)x - 2|\} \, dx \) is:
\[
2 \ln(\sqrt{2}) + \frac{\pi}{4}
\]
