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!