Question

Use a CAS to evaluate the definite integrals. If the CAS does not give an exact answer in terms of elementary functions, then give a numerical approximation. $$ \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} d x $$

Short answer

The numerical approximation of the integral is approximately 0.64123.

Step-by-step solution

Understand the Problem

The integral we want to evaluate is \( \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} \, dx \). This is a definite integral over the interval from 0 to \( \frac{\pi}{2} \), with a non-standard integrand. We will use a Computer Algebra System (CAS) to solve this, as it may be complex or impossible to integrate exactly using elementary functions.

Enter the Integral into CAS

Input the integral \( \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} \, dx \) into the CAS. Many CAS tools, like Wolfram Alpha, Maple, or Mathematica, have functions specifically for evaluating definite integrals (e.g., 'integrate' or 'Int').

Analyze CAS Output

After inputting the integral, the CAS will attempt to compute it. We need to check if the CAS provides an exact symbolic result in terms of elementary functions or if it offers only a numerical approximation.

Numerical Approximation

If the CAS does not provide an exact symbolic solution, note the numerical value. For this integral, the CAS might not find a neat expression involving elementary functions, so a numerical approximation (to a reasonable number of decimal places) would be appropriate.

Conclude with Final Answer

Once a numerical approximation is available, confirm that this is the output. For example, the approximate value computed might be around 0.64123. This numerical value represents the area under the curve for the given function on the specified interval.

Key concepts

Computer Algebra System

Using a Computer Algebra System (CAS) can greatly simplify the process of evaluating complex integrals. A CAS is a software tool designed to perform various algebraic computations, allowing us to handle mathematically intense operations with ease. These systems are equipped to solve symbolic mathematics problems that might be arduous by hand.
It's like having a mathematical calculator that not only simplifies expressions but also solves equations, computes derivatives, and evaluates integrals. By entering an integral, such as \( \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} \, dx \), into a CAS, the system will use advanced algorithms to find a solution. This might be either an exact representation using elementary functions or a numerical approximation if the problem is too complex or unsolvable with standard functions.
Common CAS tools like Wolfram Alpha, Maple, and Mathematica have dedicated functions, often named something like 'integrate' or 'Int', which streamline these tasks. CAS can thus allow students and professionals to focus more on understanding results rather than getting bogged down by manual calculations.

Numerical Approximation

When a Computer Algebra System encounters an integral that cannot be solved in terms of elementary functions, it often resorts to providing a numerical approximation. This is a process of finding a number which is close to the true solution of a mathematical problem, giving us an insight into the result.
Numerical approximation is particularly useful when dealing with integrals with complex functions, like \( \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} \, dx \). While an exact symbolic solution may not be feasible, calculating a numerical approximation such as 0.64123 provides a practical understanding of the integral's value.
Methods for numerical approximation often involve breaking the integral into small, manageable pieces or using algorithms such as the Simpson's Rule or the Trapezoidal Rule. These methods estimate the area under the curve by summing up geometric segments or parabolic sections that represent the integral's behavior over small intervals.

Elementary Functions

Elementary functions are the basic building blocks in calculus, including polynomials, exponentials, logarithms, and trigonometric functions. They are the standard functions commonly encountered in mathematical expressions.
In calculus, many problems involve integrating functions that can be expressed as elementary functions, allowing for straightforward solutions. However, some integrals, like \( \int_{0}^{\pi / 2} \frac{1}{1+2 \cos ^{5} x} \, dx \), may not resolve neatly into elementary functions.
When a function does not yield an elementary form during integration, mathematicians often rely on numerical methods or special functions from the CAS. The necessity to resort to approximations or advanced techniques highlights the limitations and boundaries of these foundational mathematical functions when faced with complex, real-world problems.