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explain this math part by part very easily

The problem involves integrating a trigonometric function, and the solution is broken down into steps. Let’s go through each part step by step.

### Problem Statement
The goal is to integrate the function \(\int \sin^3(x) \, dx\).

### Step 1: Simplification using Identity

The solution starts with:
\[
\int \sin^3(x) \, dx
\]

The function \(\sin^3(x)\) can be rewritten using trigonometric identities. Specifically, the identity used is:
\[
\sin^3(x) = \sin(x) \cdot \sin^2(x)
\]
And since \(\sin^2(x)\) can be expressed as:
\[
\sin^2(x) = 1 - \cos^2(x)
\]
Thus:
\[
\sin^3(x) = \sin(x) \cdot (1 - \cos^2(x))
\]
So the integral becomes:
\[
\int \sin(x)(1 - \cos^2(x)) \, dx
\]

### Step 2: Substitution

Next, the substitution \( u = \cos(x) \) is applied. This means:
\[
du = -\sin(x) \, dx
\]
Therefore, the integral changes to:
\[
-\int (1 - u^2) \, du
\]
This integral is easier to solve.

### Step 3: Integrating the Simplified Expression

Now, the integral is split into two parts:
\[
-\int 1 \, du + \int u^2 \, du
\]
This gives:
\[
-u + \frac{u^3}{3} + C
\]
Where \(C\) is the constant of integration.

### Step 4: Substitution Back

Finally, substitute back \( u = \cos(x) \):
\[
-\cos(x) + \frac{\cos^3(x)}{3} + C
\]
This is the final answer.

### Summary

1. **Simplification**: Break down \(\sin^3(x)\) using trigonometric identities.
2. **Substitution**: Use substitution to simplify the integral.
3. **Integration**: Solve the resulting integral.
4. **Final Answer**: Substitute back to get the final expression in terms of \(x\).

Would you like further details on any of the steps, or do you have any questions?

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### Related Questions

1. How do you derive the identity \(\sin^2(x) = 1 - \cos^2(x)\)?
2. What are the general steps for integrating a trigonometric function?
3. How does substitution simplify trigonometric integrals?
4. Can you provide an example of integrating \(\cos^3(x)\)?
5. What are some common mistakes to avoid when solving trigonometric integrals?

### Tip

Always check if a trigonometric identity can simplify the integral before starting the integration process.

This detailed guide explains how to integrate the trigonometric function sin^3(x) using trigonometric identities and substitution. Suitable for students in Grades 11-12.

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