To find the integral of $\sin^3x$, let's go through the correct process:

Step 1: Express $\sin^3x$ in terms of simpler functions.

Use the identity: $\sin^3x = \sin x \cdot \sin^2x = \sin x \cdot (1 - \cos^2x)$ Thus, $\sin^3x = \sin x - \sin x \cdot \cos^2x$

Step 2: Integrate the expression.

Now, integrate term by term: $\int \sin^3x \, dx = \int \sin x \, dx - \int \sin x \cdot \cos^2x \, dx$

Integrating $\int \sin x \, dx$:

$\int \sin x \, dx = -\cos x$

Integrating $\int \sin x \cdot \cos^2x \, dx$:

For this, use substitution. Let $u = \cos x$, hence $du = -\sin x \, dx$.

The integral becomes: $\int \sin x \cdot \cos^2x \, dx = -\int u^2 \, du = -\frac{u^3}{3} = -\frac{\cos^3x}{3}$

Step 3: Combine the results.

Putting it all together: $\int \sin^3x \, dx = -\cos x + \frac{\cos^3x}{3} + C$ where $C$ is the constant of integration.

Conclusion:

The correct integral is: $\boxed{-\cos x + \frac{\cos^3x}{3} + C}$

You mentioned that you were getting $\cos x + \frac{\cos^3x}{3}$, but there should be a negative sign in front of $\cos x$ in the final answer.

Would you like more details or have any questions? Here are some related questions you might explore:

1. How would you integrate $\cos^3x$?
2. What is the integral of $\sin^nx$ for any integer $n$?
3. How does integration by parts apply to trigonometric functions?
4. Can we use reduction formulas for $\sin^nx$ integration?
5. What are some common mistakes when integrating powers of trigonometric functions?

Tip: Always double-check the signs in trigonometric integrals, as they can easily affect the final result.