The integral of the absolute value of \(\sin(x)\), denoted by \(\int |\sin(x)| \, dx\), can be approached by breaking the function into intervals where \(\sin(x)\) is positive or negative. This is because the absolute value function affects the behavior of \(\sin(x)\) differently depending on the interval.

### Step-by-Step Solution

1. **Break the integral into intervals:**

   \(\sin(x)\) is positive on the interval \([2n\pi, (2n+1)\pi]\), and negative on the interval \([(2n+1)\pi, (2n+2)\pi]\) for integer \(n\). Thus, the absolute value of \(\sin(x)\) can be handled by splitting the integral into two parts.

2. **Positive intervals (where \(\sin(x) \geq 0\)):**
   On \([2n\pi, (2n+1)\pi]\), \(|\sin(x)| = \sin(x)\). The integral in this region is:
   \[
   \int \sin(x) \, dx = -\cos(x) + C
   \]

3. **Negative intervals (where \(\sin(x) < 0\)):**
   On \([(2n+1)\pi, (2n+2)\pi]\), \(|\sin(x)| = -\sin(x)\). The integral in this region is:
   \[
   \int -\sin(x) \, dx = \cos(x) + C
   \]

4. **General formula:**
   The integral of \(|\sin(x)|\) over a full period, from \(0\) to \(2\pi\), is the sum of the integrals over the intervals \([0, \pi]\) and \([\pi, 2\pi]\):
   \[
   \int_0^\pi \sin(x) \, dx + \int_\pi^{2\pi} -\sin(x) \, dx
   \]

   Evaluating these:
   \[
   \int_0^\pi \sin(x) \, dx = [-\cos(x)]_0^\pi = 2
   \]
   \[
   \int_\pi^{2\pi} -\sin(x) \, dx = [\cos(x)]_\pi^{2\pi} = 2
   \]

   Therefore, over a full period:
   \[
   \int_0^{2\pi} |\sin(x)| \, dx = 2 + 2 = 4
   \]

### Final Answer:
The indefinite integral of \(|\sin(x)|\) is a piecewise function, depending on the intervals where \(\sin(x)\) is positive or negative. Over one period \([0, 2\pi]\), the integral evaluates to \(4\).

Would you like more details on any step or further clarifications?

Here are some related questions:
1. What is the integral of \(\cos(x)\)?
2. How can you calculate the area under the curve of \(\sin(x)\)?
3. How do you handle integrals involving absolute values of more complex functions?
4. How does the integral of \(\sin(x)\) change over different intervals?
5. What is the Fourier series of \(|\sin(x)|\)?

**Tip:** Always check the periodicity of trigonometric functions when solving integrals, as they can help simplify the calculations.

Understand the integral of |sin(x)| with a step-by-step approach, breaking it into intervals where sin(x) is positive or negative. Learn key concepts in calculus, including piecewise integration and periodicity, with detailed formulas and explanations. Suitable for high school and college students.

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