Question

Evaluate the following integrals. \(\int_{\pi / 3}^{\pi / 2} 2 \sec (2 \theta) \tan (2 \theta) d \theta\)

Short answer

The value of the integral is \(\frac{1}{2}\).

Step-by-step solution

Recognize the Integral Formula

Identify that the integral's structure matches the derivative of the secant function. The derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \). Therefore, the integral \( \int \sec(2\theta) \tan(2\theta) d\theta \) suggests a straightforward antidifferentiation process.

Set Up the Substitution

To simplify the integration, perform a substitution: let \( u = 2\theta \). Then, \( du = 2 d\theta \), or equivalently, \( d\theta = \frac{1}{2} du \). Substitute these into the integral, yielding \( \int \sec(u) \tan(u) \frac{1}{2} du \).

Simplify the Integral

Factor out the constant \( \frac{1}{2} \) from the integral. The integral now becomes \( \frac{1}{2} \int \sec(u) \tan(u) du \).

Integrate with Respect to u

Since the integral \( \int \sec(u) \tan(u) du \) directly equates to \( \sec(u) \), the integration becomes: \( \frac{1}{2} \sec(u) + C \), where \( C \) is the integration constant.

Substitute Back for \(\theta\)

Replace \( u \) by \( 2\theta \) to reverse the substitution: \( \frac{1}{2} \sec(2\theta) \).

Evaluate the Definite Integral

To find the definite integral, evaluate from \( \theta = \frac{\pi}{3} \) to \( \theta = \frac{\pi}{2} \). Calculate \( \frac{1}{2}(\sec(\pi) - \sec(\frac{2\pi}{3})) \).

Simplify the Evaluated Result

Evaluate \( \sec(\pi) \), which equals \(-1\), and \( \sec(\frac{2\pi}{3}) \), which is equivalent to \(-2 \). Thus, \( \frac{1}{2}((-1) - (-2)) = \frac{1}{2}(1) = \frac{1}{2} \).

Key concepts

Integration Techniques

Integration is a cornerstone of calculus, crucial for finding areas, volumes, and central points. Various integration techniques make tackling complex integrals easier. In this problem, we focus on using substitution and recognizing patterns to ease the process. Recognizing that the integrand resembles the derivative of the secant function saves us effort.

• Substitution: This technique, also known as the change of variables, transforms an integral into a simpler form. Substitution is particularly useful when dealing with integrals of composite functions.
• Pattern Recognition: Spotting patterns in integrals can guide us to their antiderivatives quickly. In our exercise, identifying that the integrand matched the derivative of the secant helped streamline the integration.

Understanding these techniques and knowing how to apply them effectively is key to mastering more complex integrals.

Definite Integral

The definite integral is an essential concept in calculus, representing the signed area under a curve between two points. It combines the process of finding an antiderivative with an evaluation at upper and lower limits. In this problem, the definite integral \[ \ \int_{\pi / 3}^{\pi / 2} 2 \sec (2 \theta) \tan (2 \theta) d \theta \] requires us to compute the antiderivative and evaluate it at the boundaries.

• Limits of Integration: These are the boundary points, \( \theta = \pi / 3 \) and \( \theta = \pi / 2 \) , that provide our evaluation range.
• Evaluation: After integrating, evaluating at these limits involves plugging them into the antiderivative and subtracting the lower limit outcome from the upper one.
• Simplification: Finally, simplify the expression to extract a numerical result reflecting the net area under the curve.

These steps confirm that our integral accurately reflects real-world quantities, such as areas and net changes.

Trigonometric Functions

Trigonometric functions like \( \sec \theta \) and \( \tan \theta \) are fundamental in calculus, occurring frequently in integrals, derivatives, and real-world applications.

• Secant Function: Defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \), this function's derivative is \( \sec(\theta) \tan(\theta) \). Recognizing this relationship was key in solving the integral.
• Tangent Function: Defined as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), it's often paired with secant in integration problems.

These functions not only have geometric interpretations, depicting ratios in right triangles, but also have wide-reaching implications in physics, engineering, and beyond. Understanding their relationships and how they interact with calculus principles equips you to handle various calculations efficiently.