Problem 93 - Precalculus

Question

$$\int_{7 \pi / 6}^{5 \pi / 4} \sec ^{2} x d x$$

Step-by-Step Solution

Answer

The integral evaluates to \(1 - \frac{\sqrt{3}}{3}\).

Step 1: Identify the Integral Form

The integral we are dealing with is \( \int \sec^2 x \, dx \). The standard integral formula for \( \sec^2 x \) is its antiderivative, \( \tan x \).

Step 2: Apply the Integral Formula

Using the antiderivative, we have: \( \int \sec^2 x \, dx = \tan x + C \). For a definite integral from \( a \) to \( b \), it becomes \( [\tan x]_a^b = \tan b - \tan a \).

Step 3: Evaluate the Antiderivative at the Upper Limit

Evaluate \( \tan x \) at the upper limit: \( x = \frac{5\pi}{4} \). Since \( \tan \left( \frac{5\pi}{4} \right) = 1 \).

Step 4: Evaluate the Antiderivative at the Lower Limit

Evaluate \( \tan x \) at the lower limit: \( x = \frac{7\pi}{6} \). Since \( \tan \left( \frac{7\pi}{6} \right) = \frac{1}{\sqrt{3}} \).

Step 5: Subtract the Evaluated Values

The value of the definite integral is \( \tan \left( \frac{5\pi}{4} \right) - \tan \left( \frac{7\pi}{6} \right) = 1 - \frac{1}{\sqrt{3}} \). Simplifying further, we can rationalize to get \( 1 - \frac{\sqrt{3}}{3} \).

Step 6: Conclusion

Thus, the value of the integral \( \int_{7\pi/6}^{5\pi/4} \sec^2 x \, dx \) is \( 1 - \frac{\sqrt{3}}{3} \).

Key Concepts

Understanding the Antiderivative

The concept of an antiderivative is crucial in calculus because it provides the foundation for solving integrals. In simple terms, the antiderivative of a function is another function that, when differentiated, gives the original function back. For instance, the antiderivative of the function \( \sec^2 x \) is \( \tan x \).
This is because the derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \). It's like working backwards from what you find in derivative problems.

• The process of finding an antiderivative is known as integration.
• If \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).
• Antiderivatives are essential for solving definite integrals.

Understanding this concept helps you unlock the mysteries of definite integrals and makes trigonometric integrals more approachable.

Exploring Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions. These types of integrals are common in calculus due to the periodic nature of trigonometric functions, which are applicable in various fields such as physics and engineering.
For our particular problem, we are working with the integral of \( \sec^2 x \). This requires knowledge of specific integral formulas, such as knowing that the integral of \( \sec^2 x \) is \( \tan x \).

• Integral formulas for trigonometric functions can simplify complex problems.
• Common trigonometric integrals include those involving \( \sin x \), \( \cos x \), and \( \tan x \).
• Familiarity with these integral formulas is essential to solve or simplify trigonometric integrals effectively.

By understanding these formulas, you can confidently tackle trigonometric integrals and apply them to definite integrals.

The Art of Evaluating Limits in Definite Integrals

Evaluating limits is an integral part of solving definite integrals. The term "limits" here refers to the boundaries of integration, \( a \) and \( b \), which specify the interval over which the function is integrated.
In our exercise, we calculate the definite integral from \( 7\pi/6 \) to \( 5\pi/4 \) of \( \sec^2 x \) by substituting these limits into the antiderivative \( \tan x \).

• First, evaluate the antiderivative at the upper limit, \( b \).
• Next, evaluate the antiderivative at the lower limit, \( a \).
• Finally, subtract the result of the lower limit from the upper limit to find the value of the integral.

This process, using the fundamental theorem of calculus, transforms an indefinite integral into a calculable quantity. Understanding this approach allows you to find the specific area under a curve defined by trigonometric functions within specified limits.