Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             
                                                                                                                                                                    
 ∫ 𝓍 cos²𝓍² d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves a trigonometric function squared, specifically cos²(𝓍²). To simplify, use the trigonometric identity cos²(u) = (1 + cos(2u)) / 2, where u = 𝓍² in this case.
2. Step 2: Substitute the identity into the integral. The integral becomes ∫ 𝓍 * (1 + cos(2𝓍²)) / 2 d𝓍. Split the integral into two parts: ∫ 𝓍/2 d𝓍 + ∫ 𝓍 * cos(2𝓍²)/2 d𝓍.
3. Step 3: For the first term, ∫ 𝓍/2 d𝓍, integrate directly using the power rule for integration: ∫ 𝓍^n dx = (𝓍^(n+1)) / (n+1). This gives (𝓍² / 4).
4. Step 4: For the second term, ∫ 𝓍 * cos(2𝓍²)/2 d𝓍, use substitution. Let u = 2𝓍², so du = 4𝓍 d𝓍. Rewrite the integral in terms of u: (1/8) ∫ cos(u) du. The integral of cos(u) is sin(u), so this term becomes (1/8) sin(2𝓍²).
5. Step 5: Combine the results from both terms. The final expression for the integral is (𝓍² / 4) + (1/8) sin(2𝓍²) + 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 sin²x + cos²x = 1, and double angle formulas. These identities are essential for simplifying integrals involving sin²x and cos²x, allowing for easier evaluation.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving trigonometric functions like cos²x, recognizing patterns and applying appropriate techniques is crucial for finding the antiderivative.

Definite and Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions without limits. Understanding the difference is important when evaluating integrals, as it affects the final result. In this context, knowing whether to apply limits or find a general antiderivative is key to solving the problem.