3. Describe the method used to integrate sin³x.

Verified step by step guidance

1. To integrate \( \sin^3(x) \), start by recognizing that this is a power of sine function. A common approach is to use a trigonometric identity to simplify the expression. Specifically, split \( \sin^3(x) \) into \( \sin(x) \cdot \sin^2(x) \).
2. Next, use the Pythagorean identity \( \sin^2(x) = 1 - \cos^2(x) \) to rewrite \( \sin^2(x) \). This gives \( \sin^3(x) = \sin(x) \cdot (1 - \cos^2(x)) \).
3. Now, let \( u = \cos(x) \). Then, \( du = -\sin(x) \, dx \). Substitute \( \sin(x) \, dx \) with \( -du \) and \( \cos(x) \) with \( u \). The integral becomes \( -\int (1 - u^2) \, du \).
4. Break the integral into two simpler parts: \( -\int 1 \, du + \int u^2 \, du \). Integrate each term separately using basic power rule for integration.
5. Finally, after integrating, substitute back \( u = \cos(x) \) to return to the original variable \( x \). Simplify the result to complete the solution.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. For sin³x, one common technique is to use trigonometric identities and substitution to simplify the integrand. Understanding these techniques is essential for solving integrals that involve powers of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. For sin³x, the identity sin²x = 1 - cos²x can be used to rewrite the integrand, making it easier to integrate. Familiarity with these identities is crucial for manipulating trigonometric expressions in integration.

Substitution Method

The substitution method is a technique used in integration where a new variable is introduced to simplify the integral. In the case of sin³x, substituting u = cosx can transform the integral into a more manageable form. This method is particularly useful when dealing with composite functions or powers of trigonometric functions.