Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

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. $$ \int_{0}^{\pi} \cos ^{4} \frac{x}{2} d x $$

Step-by-Step Solution

Answer

The numerical approximation using CAS is approximately 1.1781.

Step 1: Rewrite the Integral in Terms of a CAS-Solvable Function

The given integral is \( \int_{0}^{\pi} \cos ^{4} \frac{x}{2} \, dx \). This involves a power of cosine, which can be expanded using a trigonometric identity. The identity for \( \cos^2 \theta \) is \( \frac{1 + \cos(2\theta)}{2} \). First, rewrite \( \cos^4 \frac{x}{2} \) as \( \left( \cos^2 \frac{x}{2} \right)^2 = \left( \frac{1 + \cos x}{2} \right)^2 \).

Step 2: Expand the Expression

Expand \( \left( \frac{1 + \cos x}{2} \right)^2 \) as follows:\( \frac{1 + \cos x}{2} \times \frac{1 + \cos x}{2} = \frac{(1 + \cos x)^2}{4} = \frac{1 + 2\cos x + \cos^2 x}{4} \).

Step 3: Apply the Double Angle Identity to \( \cos^2 x \)

Now, use the identity for \( \cos^2 x \), which is \( \cos^2 x = \frac{1 + \cos(2x)}{2} \). Substitute this into the expression:\( \frac{1 + 2\cos x + \frac{1 + \cos(2x)}{2}}{4} \).

Step 4: Simplify the Expression

Combine terms:\( \frac{4 + 4\cos x + 1 + \cos(2x)}{8} = \frac{5 + 4\cos x + \cos(2x)}{8} \). This simplifies the integrand to a form that uses basic trigonometric and algebraic expressions.

Step 5: Evaluate the Integral Using CAS

Input the integral \( \int_{0}^{\pi} \frac{5 + 4\cos x + \cos(2x)}{8} \, dx \) into a CAS. The CAS uses techniques such as substitution and numerical integration to find the value of this definite integral.

Step 6: Check CAS Result

After inputting the function into the CAS, verify whether an exact or numerical result is provided. In this case, a numerical approximation is often given as the integral of certain trigonometric functions over a period can result in non-elementary answers.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that simplify expressions involving trigonometric functions. These identities are essential tools when working with integrals of trigonometric functions, particularly when those functions are raised to a power.
A classic example is the power reduction identity. It helps transform powers of sines and cosines into simpler forms that are easier to integrate. For instance, in the exercise we are dealing with, the expression \( \cos^4 \frac{x}{2} \) can be rewritten using the identity \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \).

• By applying this identity once, \( \cos^2 \theta \) is transformed into a sum of elementary trigonometric functions.
• Applying the identity again to \( \cos^4 \theta \) results in a much simpler expression.

These identities not only ease the workload but also transition a complex expression into a straightforward format suitable for further mathematical manipulation.

Numerical Approximation

Numerical approximation methods play a crucial role when exact answers in terms of elementary functions are not possible. Often in calculus, especially with trigonometric integrals over specific intervals, the result might not be a neat expression.
In such scenarios, numerical approximation techniques offer viable solutions. Techniques like the trapezoidal rule, Simpson's rule, or using a computer algebra system (CAS) to perform numerical integration are commonly used.
These methods work by breaking the interval into smaller subintervals:

• The integral over these small subintervals is approximated using simpler functions (e.g., linear or quadratic).
• The results are accumulated to get a close estimate of the integral’s value over the entire interval.

Although these aren't exact, the approximations provide insights where algebraic solutions are infeasible.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a robust software tool adept at performing symbolic and numerical computations. These systems are incredibly useful in calculus for evaluating complex integrals, like the one in the given exercise.
With CAS, users input functions and request calculations such as integrations, derivatives, or simplifications. Here are some typical steps CAS performs:

• Translating mathematical expressions into a computer-readable form.
• Using algorithmic processes to attempt solving the problem symbolically first.
• When symbolic solutions are too complex or impossible, CAS switches to numerical approximations.

CAS not only speeds up computation but also circumvents human error, thus ensuring accurate and reliable results. It empowers students and professionals alike in tackling advanced calculus problems efficiently.