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Chapter 8: Problem 44

Evaluate the definite integral. Use a symbolic integration utility to verify your results. $$ \int_{0}^{\pi / 8} \sin 2 x \cos 2 x d x $$

Short Answer

Expert verified
The value of the definite integral is 1/8.

Step by step solution

01

Recognize the integrand as half of the double-angle formula

Identify the integral as a form of the Double-Angle Formula for sine: \( \sin 2x = 2 \sin x \cos x \). Given the integrand is \( \sin 2x \cos 2x \), it can be expressed as \(1/2 \sin 4x\). So, \( \int_{0}^{\pi / 8} \sin 2 x \cos 2 x d x \) becomes \( \int_{0}^{\pi / 8}1/2 \sin 4x d x \).
02

Perform the integration using antiderivatives

Now integrate \( \sin 4x \) over the interval [0, \( \pi / 8 \)]. The antiderivative of \( \sin 4x \) is \( -1/4 \cos 4x \). Incorporating the 1/2 in front gives \( -1/8 \cos 4x \). So, the integral becomes \( -1/8[ \cos (4x)]_{0}^{\pi / 8} \).
03

Evaluate the definite integral

Evaluate the integral by substituting \( x = \pi / 8 \) and \( x = 0 \) into \( -1/8[ \cos (4x)] \). This gives \( -1/8[ \cos (\pi / 2) - \cos (0)] \), which simplifies to \( -1/8[0 - 1] = 1/8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double-Angle Formula
The double-angle formula is a trigonometric identity that expresses trigonometric functions of double angles in terms of single angles. It's particularly useful in simplifying expressions and solving integrals that involve trigonometric functions at double angles.
  • One common form of the double-angle formula is for sine: \( \sin 2x = 2 \sin x \cos x \).
  • This identity helps in expressing products of sine and cosine functions in simpler forms.
In the context of our exercise, the original integral \( \int \sin 2x \cos 2x \, dx \) is restructured as \( \frac{1}{2} \sin 4x \) using the double-angle formula. This simplification makes the integral easier to solve. Understanding these formulas is key to tackling integrals involving trigonometric functions.
Symbolic Integration
Symbolic integration is a method of finding the integral of a function symbolically, rather than numerically. It involves finding an expression in terms of symbols that represents the antiderivative or integral of a given function.
  • This approach is fundamental in calculus and helps ensure the results are as exact as possible.
  • Unlike numerical integration, symbolic integration derives an exact algebraic expression.
In our problem, symbolic integration is used to evaluate \( \int \frac{1}{2} \sin 4x \, dx \), resulting in the expression \( -\frac{1}{8} \cos 4x \). By understanding symbolic integration, one can manually check against results obtained from symbolic calculators like those in computational software.
Antiderivative
An antiderivative is a function whose derivative is the given function. In the context of integrals, an antiderivative represents the indefinite integral of a function.
  • Finding the antiderivative is a crucial step in the process of integration.
  • The antiderivative for \( \sin 4x \) is \( -\frac{1}{4} \cos 4x \).
By taking into account any constant multiplier, such as \( \frac{1}{2} \) in our example, the complete integration results in \( -\frac{1}{8} \cos 4x \). Knowing how to find and use antiderivatives is essential for evaluating definite integrals.
Evaluation of Integrals
The evaluation of integrals involves calculating the definite integral by applying limits to the antiderivative, which provides a numerical result.
  • Definite integrals are computed over specified intervals and yield a precise numerical value.
  • To evaluate them, one substitutes the boundary values into the antiderivative.
In the exercise, substituting the limits \( x = \frac{\pi}{8} \) and \( x = 0 \) into \( -\frac{1}{8} \cos 4x \) results in an evaluated integral of \( \frac{1}{8} \). This step shows how the area under the curve is computed within defined bounds, connecting symbolic integration with practical applications.

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