Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             
                                                                                                                                                                    
 ∫ sin² 𝓍 d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves sin²(𝓍). To simplify this, use the trigonometric identity sin²(𝓍) = (1 - cos(2𝓍)) / 2.
2. Step 2: Rewrite the integral using the identity: ∫ sin²(𝓍) d𝓍 = ∫ [(1 - cos(2𝓍)) / 2] d𝓍.
3. Step 3: Split the integral into two parts: ∫ [(1/2) - (cos(2𝓍)/2)] d𝓍 = (1/2) ∫ 1 d𝓍 - (1/2) ∫ cos(2𝓍) d𝓍.
4. Step 4: Evaluate each part separately. For the first term, ∫ 1 d𝓍 = 𝓍. For the second term, use the formula for the integral of cos(k𝓍): ∫ cos(k𝓍) d𝓍 = (1/k) sin(k𝓍). Here, k = 2.
5. Step 5: Combine the results: (1/2)𝓍 - (1/4)sin(2𝓍) + C, where C is the constant of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, such as sin²(x) + cos²(x) = 1, which can simplify integrals involving sin²(x) and cos²(x). Understanding these identities is crucial for transforming integrals into more manageable forms.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and using trigonometric identities. For integrals involving sin²(x), the power-reduction formula can be particularly useful, allowing the integral to be expressed in terms of simpler functions that are easier to integrate.

Definite and Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is essential when evaluating integrals, as it affects the final result and the interpretation of the integral in a given context.