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?

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.