49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves cos⁶(x), which is a power of a trigonometric function. To simplify, use the power-reduction formula for cosine: cos²(x) = (1 + cos(2x)) / 2. This will help reduce the power of cosine in the integrand.
2. Step 2: Rewrite cos⁶(x) as (cos²(x))³ and apply the power-reduction formula to cos²(x). Substitute cos²(x) = (1 + cos(2x)) / 2 into the expression, resulting in cos⁶(x) = [(1 + cos(2x)) / 2]³.
3. Step 3: Expand [(1 + cos(2x)) / 2]³ using the binomial theorem. This will give you terms involving powers of cos(2x). The expanded form will include terms like 1, cos(2x), cos²(2x), and cos³(2x).
4. Step 4: Integrate each term separately over the interval [0, π/2]. For terms involving cos²(2x) and cos³(2x), use appropriate trigonometric identities or substitution methods to simplify the integration process. For example, cos²(2x) can be reduced using the power-reduction formula again.
5. Step 5: Use a computer algebra system (CAS) to evaluate the integral for both the exact result and the approximate numerical result. Input the integral ∫ from 0 to π/2 of cos⁶(x) dx into the CAS, and it will compute the values for you.

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, over the interval from the lower limit to the upper limit.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is software designed to perform symbolic mathematics, allowing users to manipulate mathematical expressions in a way similar to traditional algebra. CAS can evaluate integrals, solve equations, and simplify expressions, providing both exact and numerical results. This technology is particularly useful for complex integrals that are difficult to solve analytically.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions that include trigonometric functions, such as sine and cosine. These integrals often require specific techniques, such as substitution or the use of trigonometric identities, to simplify the integrand. In the case of the integral ∫ cos⁶x dx, recognizing patterns and applying reduction formulas can facilitate the evaluation process.