Evaluating integrals Evaluate the following integrals.                                                                                                                                         
                                                                                                                                                                    
 ∫ 𝓍 sin 𝓍² cos⁸ 𝓍² d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves a composite function. The term sin(𝓍²) and cos⁸(𝓍²) suggest that substitution might simplify the problem. Let u = 𝓍², so that du = 2𝓍 d𝓍.
2. Step 2: Rewrite the integral using the substitution u = 𝓍². This transforms the integral into (1/2) ∫ sin(u) cos⁸(u) du, where the factor of 1/2 comes from adjusting for the differential du = 2𝓍 d𝓍.
3. Step 3: Simplify the integral further. Notice that cos⁸(u) can be expressed as (cos²(u))⁴. Use the Pythagorean identity cos²(u) = 1 - sin²(u) to rewrite cos⁸(u) in terms of sin(u).
4. Step 4: Expand the expression (1 - sin²(u))⁴ using the binomial theorem. This will result in a polynomial in terms of sin(u), which can be integrated term by term.
5. Step 5: Integrate each term of the expanded polynomial with respect to u. After completing the integration, substitute back u = 𝓍² to return to the original variable 𝓍.

Key Concepts

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and trigonometric identities. Understanding these techniques is crucial for solving complex integrals, as they allow for the simplification of the integrand into a more manageable form. For example, substitution can transform an integral into a simpler variable, making it easier to evaluate.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially when dealing with integrals involving these functions. In the given integral, the presence of sin(x²) and cos(x²) suggests that trigonometric identities or substitutions may be necessary to simplify the expression. Familiarity with the properties and graphs of these functions aids in understanding their behavior during integration.

Definite vs. Indefinite Integrals

Integrals can be classified as definite or indefinite, with the former having specific limits of integration and the latter not. The integral in the question is indefinite, meaning it represents a family of functions rather than a numerical value. Recognizing this distinction is important for correctly interpreting the results of integration and understanding the context in which the integral is applied.