https://file.aiquickdraw.com/mathbot/file/172779511798121utq2om.jpg

The problem displayed in the image is an integration problem:

\[
\int \sin^3(x) \, dx
\]

We need to solve this integral and match it to one of the given options.

### Step 1: Use trigonometric identity
We start by rewriting \( \sin^3(x) \) using a trigonometric identity:
\[
\sin^3(x) = \sin(x) \cdot \sin^2(x)
\]
Since \( \sin^2(x) = 1 - \cos^2(x) \), we can rewrite the expression as:
\[
\sin^3(x) = \sin(x)(1 - \cos^2(x)) = \sin(x) - \sin(x) \cos^2(x)
\]

### Step 2: Split the integral
Now we split the original integral:
\[
\int \sin^3(x) \, dx = \int \left( \sin(x) - \sin(x) \cos^2(x) \right) dx
\]
This can be broken into two separate integrals:
\[
\int \sin(x) \, dx - \int \sin(x) \cos^2(x) \, dx
\]

### Step 3: Solve the first integral
The first integral is straightforward:
\[
\int \sin(x) \, dx = -\cos(x)
\]

### Step 4: Solve the second integral using substitution
For the second integral, we use the substitution:
\[
u = \cos(x) \quad \Rightarrow \quad du = -\sin(x) \, dx
\]
Thus, the second integral becomes:
\[
-\int u^2 \, du = -\frac{u^3}{3} = -\frac{\cos^3(x)}{3}
\]

### Step 5: Combine the results
Now, combining both results:
\[
\int \sin^3(x) \, dx = -\cos(x) + \frac{\cos^3(x)}{3} + C
\]

### Step 6: Match the options
This matches the first option:
\[
\boxed{-\frac{\cos^3(x)}{3} - \cos(x) + C}
\]

Would you like further clarification on any step? Here are five related questions for practice:

1. How do you integrate \( \cos^3(x) \, dx \)?
2. What is the integral of \( \sin^4(x) \, dx \)?
3. How do you use substitution for \( \int \sin(x) \cos^2(x) \, dx \)?
4. Can you explain the trigonometric identity \( \sin^2(x) = 1 - \cos^2(x) \) in more detail?
5. What are common substitution methods for integrals involving trigonometric functions?

**Tip:** When dealing with odd powers of sine or cosine, break the power into smaller parts and use trigonometric identities!

Learn how to solve the integral ∫ sin^3(x) dx using trigonometric identities and substitution methods. This problem involves breaking the sine cube into simpler parts and applying key integration techniques. Ideal for university-level calculus students.

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