Problem 9 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-14\) $$ \int_{-\pi / 4}^{\pi / 4} 16 \sin ^{2} x \cos ^{2} x d x $$

Step-by-Step Solution

Answer

The value of the integral is \(\pi\).

Step 1: Simplify the integrand using trigonometric identity

First, use the trigonometric identity for \( \sin^2 x \cos^2 x \): \( \sin^2 x \cos^2 x = \frac{1}{4}\sin^2(2x) \). This helps us to write the integral in a simpler form. Thus, the integral becomes \[ \int_{-\pi / 4}^{\pi / 4} 16 \sin ^{2} x \cos ^{2} x \, dx = 16 \int_{-\pi / 4}^{\pi / 4} \frac{1}{4} \sin^2(2x) \, dx = 4 \int_{-\pi / 4}^{\pi / 4} \sin^2(2x) \, dx. \]

Step 2: Use a power-reducing identity

Use the power-reducing identity for \( \sin^2(2x) \): \( \sin^2(2x) = \frac{1 - \cos(4x)}{2} \). Rewrite the integral with this identity: \[ 4 \int_{-\pi / 4}^{\pi / 4} \frac{1 - \cos(4x)}{2} \, dx = 2 \int_{-\pi / 4}^{\pi / 4} (1 - \cos(4x)) \, dx. \]

Step 3: Split the integral

Now, split the integral into two separate integrals: \[ 2 \int_{-\pi / 4}^{\pi / 4} (1 - \cos(4x)) \, dx = 2 \left( \int_{-\pi / 4}^{\pi / 4} 1 \, dx - \int_{-\pi / 4}^{\pi / 4} \cos(4x) \, dx \right). \]

Step 4: Evaluate the first integral

The integral \( \int_{-\pi / 4}^{\pi / 4} 1 \, dx \) is straightforward. Since it is the integral of a constant, it evaluates to the length of the interval times the constant: \[ \int_{-\pi / 4}^{\pi / 4} 1 \, dx = \left. x \right|_{-\pi / 4}^{\pi / 4} = \frac{\pi}{4} - \left( -\frac{\pi}{4} \right) = \frac{\pi}{2}. \]

Step 5: Evaluate the second integral

Now, evaluate the integral of \( \cos(4x) \). The antiderivative of \( \cos(kx) \) is \( \frac{1}{k} \sin(kx) \). Therefore, \[ \int \cos(4x) \, dx = \frac{1}{4} \sin(4x). \] Substituting the limits, we have: \[ \left. \frac{1}{4} \sin(4x) \right|_{-\pi/4}^{\pi/4} = \frac{1}{4} (\sin(\pi) - \sin(-\pi)) = \frac{1}{4} (0 - 0) = 0. \]

Step 6: Combine and simplify

Now combine the results of the integrals: \[ 2 \left( \frac{\pi}{2} - 0 \right) = 2 \cdot \frac{\pi}{2} = \pi. \] Thus, the value of the original integral is \( \pi \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between trigonometric functions. These identities help simplify complex trigonometric expressions into more manageable forms. In the given problem, the identity used is \( \sin^2 x \cos^2 x = \frac{1}{4}\sin^2(2x) \).
This process allows us to convert the integrand into an equivalent expression that is easier to integrate. Understanding these identities equips you to transform integrals involving trigonometric functions and is crucial for efficiently solving these kinds of calculus problems.
Some commonly used trigonometric identities include:

• Pythagorean identities, such as \( \sin^2 x + \cos^2 x = 1 \).
• Angle sum and difference identities like \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \).

These are essential tools in both solving integrals and simplifying trigonometric expressions.

Power-Reducing Identity

Power-reducing identities simplify powers of trigonometric functions, making integrals easier to tackle. In solving the exercise, the power-reducing identity \( \sin^2(2x) = \frac{1 - \cos(4x)}{2} \) was employed. This identity reduces a square of a sine function to an expression involving no squares, which is beneficial when integrating.
This identity helps convert the integrand into a form that involves simpler functions. Instead of dealing with squares of trig functions, which might be complex to integrate directly, the terms are broken down accordingly, facilitating easier computation.
Understanding how to use power-reducing identities effectively allows you to simplify expressions and tackle even difficult integrals efficiently, especially those involving higher powers of trigonometric functions.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are the reverse process of differentiation. They are functions whose derivative is the given function. In the context of definite integrals, finding the antiderivative is a crucial step to evaluate the integral over a specified interval.
For instance, in the problem, the antiderivative of \( \cos(4x) \) is needed. The general rule is that the antiderivative of \( \cos(kx) \) is \( \frac{1}{k} \sin(kx) \). By using this rule, the integral \( \int \cos(4x) \, dx \) becomes \( \frac{1}{4} \sin(4x) \).
Finding antiderivatives requires you to be familiar with basic integration rules and techniques, which are foundational skills in calculus. These antiderivatives allow us to compute definite integrals by evaluating these expressions at the integration bounds.

Integration Bounds

Integration bounds define the interval over which the definite integral is evaluated. They're crucial as they determine the starting and ending points of integration.
In the given exercise, the bounds are from \( -\pi/4 \) to \( \pi/4 \). These values are used to evaluate the antiderivative computed in the integration process. Specifically, after finding an antiderivative, we substitute the upper and lower bounds into it and compute the difference, yielding the value of the definite integral.
The expression given, \( \left. x \right|_{-\pi / 4}^{\pi / 4} \), simplifies to evaluating at both \( x = \pi / 4 \) and \( x = -\pi / 4 \), and taking their difference. Understanding how integration bounds operate is key to effectively solving definite integrals as it influences the result of the integration heavily.