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.
58. ∫₀^{2π} dt / (4 + 2 sin t)²

Verified step by step guidance

1. Identify the integral to be evaluated: \(\displaystyle \int_0^{2\pi} \frac{dt}{(4 + 2 \sin t)^2}\).
2. Recognize that this is a definite integral over one full period of the sine function, which suggests symmetry or periodicity might simplify the problem.
3. Rewrite the denominator to factor out constants if helpful: \$4 + 2 \sin t = 2(2 + \sin t)\(, so the integral becomes \)\int_0^{2\pi} \frac{dt}{4 (2 + \sin t)^2}$.
4. Use a substitution or a known integral formula involving \(\sin t\) in the denominator, or apply a computer algebra system (CAS) to find the exact antiderivative or directly compute the definite integral.
5. After obtaining the exact result from the CAS, use it to find an approximate numerical value, ensuring to interpret the result in terms of the parameter \(a\) if needed.

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two limits, providing a numerical value. It is expressed as ∫_a^b f(x) dx, where a and b are the bounds. Understanding how to interpret and evaluate definite integrals is essential for solving problems involving accumulation or total change.

Integration Techniques for Trigonometric Functions

Integrals involving trigonometric functions often require specific methods such as substitution, trigonometric identities, or partial fractions. Recognizing patterns like sin² t or expressions involving (4 + 2 sin t)² helps simplify the integral before evaluation. Mastery of these techniques aids in finding exact antiderivatives.

Use of Computer Algebra Systems (CAS)

A CAS is software that performs symbolic mathematics, including integration. It can provide exact symbolic results and numerical approximations for definite integrals, especially when manual integration is complex. Using CAS tools efficiently requires understanding their input syntax and interpreting their output.