To integrate \((\sin x)^3\), we can use a trigonometric identity and substitution. Here's the step-by-step solution:

### Step 1: Use the trigonometric identity
We begin by using the identity:
\[
\sin^3 x = \sin x (1 - \cos^2 x)
\]
So the integral becomes:
\[
\int \sin^3 x \, dx = \int \sin x (1 - \cos^2 x) \, dx
\]

### Step 2: Substitution
Let \( u = \cos x \), which implies \( du = -\sin x \, dx \). The integral becomes:
\[
\int \sin x (1 - \cos^2 x) \, dx = -\int (1 - u^2) \, du
\]

### Step 3: Simplify and integrate
Now, integrate the expression:
\[
-\int (1 - u^2) \, du = -\left(u - \frac{u^3}{3}\right) + C
\]

### Step 4: Substitute back \( u = \cos x \)
Now, substitute back \( u = \cos x \):
\[
-(\cos x - \frac{\cos^3 x}{3}) + C
\]
which simplifies to:
\[
\frac{\cos^3 x}{3} - \cos x + C
\]

Thus, the integral of \((\sin x)^3\) is:
\[
\int \sin^3 x \, dx = \frac{\cos^3 x}{3} - \cos x + C
\]

Would you like more details on any step or have any questions?

Here are 5 related questions to extend your understanding:
1. How would you integrate \(\sin^5 x\)?
2. What if you were to integrate \((\cos x)^3\)?
3. How can you use a similar method to integrate \((\sin x)^2\)?
4. What is the integral of \((\sin x)^n\) for general \(n\)?
5. Can you derive the reduction formula for \(\int (\sin x)^n dx\)?

**Tip:** Trigonometric identities are very useful for simplifying integrals involving powers of sine and cosine!

Learn how to integrate (sin x)^3 using trigonometric identities and substitution in a step-by-step approach. This solution covers core trigonometry and integration techniques, suitable for high school students in Grades 11-12.

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