Problem 16 - Calculus Single Variable

Question

In Exercises \(11-20,\) calculate \(F(x)=\int_{a}^{x} f(t) d t\) $$ f(t)=\sec ^{2}(t) \quad a=0 \quad(-\pi / 2

Step-by-Step Solution

Answer

\( F(x) = \tan(x) \).

Step 1: Understand the Problem

We are given the integral function \( F(x) = \int_{a}^{x} f(t) \, dt \), where the function \( f(t) = \sec^2(t) \). Additionally, we are provided the lower limit of integration \( a = 0 \) and the domain constraint \( -\pi/2 < x < \pi/2 \). Our goal is to find \( F(x) \).

Step 2: Recall Integration Formula

The integral of \( \sec^2(t) \) with respect to \( t \) is known to be \( \tan(t) + C \), where \( C \) is a constant of integration. This is a standard result from integration involving trigonometric functions.

Step 3: Set Up the Integral

Using the integration results from the previous step, we set up the integral as \[ F(x) = \int_{0}^{x} \sec^2(t) \, dt. \]

Step 4: Solve the Definite Integral

Evaluate the definite integral \[ F(x) = \tan(t) \bigg|_{0}^{x} = \tan(x) - \tan(0). \] Since \( \tan(0) = 0 \), this simplifies to \( F(x) = \tan(x) \).

Step 5: State the Final Solution

The function \( F(x) \) is \( \tan(x) \), evaluated within the interval \( -\pi/2 < x < \pi/2 \). Therefore, \( F(x) = \tan(x) \) for the specified domain.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions, such as sine, cosine, tangent, and secant. These types of integrals are prevalent in calculus due to their applicability in various mathematical and real-world problems.
Consider the function \( f(t) = \sec^2(t) \). This is a classic example in trigonometric integrals where the integral leads to another trigonometric function, namely the tangent function.
To integrate \( \sec^2(t) \), it's useful to recall that its antiderivative is \( \tan(t) \). This conclusion is drawn from the derivative of the tangent, which is precisely \( \sec^2(t) \).

• Recognize trigonometric identities to simplify integrals.
• Use substitution if needed to match standard forms.
• Always double-check results with basic derivative rules.

Understanding these trigonometric integral relationships is key for solving problems efficiently.

Integration Techniques

Learning effective integration techniques is essential in calculus, especially when dealing with definite integrals. The function \( f(t) = \sec^2(t) \) is straightforward because it corresponds directly with a fundamental integration rule.
Here are some integration techniques to help solve similar problems:

• **Basic Antiderivatives:** Knowing the standard results, such as \( \int \sec^2(t) \, dt = \tan(t) \), to quickly evaluate integrals.
• **Substitution Method:** For complex integrals, substitution can be used to simplify the integrand into a more familiar form.
• **Integration by Parts:** Useful where integrands are products of functions; however, not necessary in this case.

In our problem, using the fundamental antiderivative rule directly gives us the result \( \tan(t) \). No need to apply more complex techniques here as the problem remains simple.

Definite Integration

Definite integration calculates the accumulation of a function's values over a specific interval. It involves integrating the function and then applying limits to find the net area under the curve.
In the given example, we calculate \( F(x) = \int_{0}^{x} \sec^2(t) \, dt \), representing the accumulated change of the function \( \sec^2(t) \) from 0 to \( x \).
Steps to solve include:

• **Evaluate Antiderivative:** Find \( \tan(t) \) as the antiderivative of \( \sec^2(t) \).
• **Apply Limits:** Substitute upper and lower limits of integration. Here, from 0 to \( x \), which simplifies to \( \tan(x) - \tan(0) \).
• **Simplify:** Knowing \( \tan(0) = 0 \) yields \( \tan(x) \) as the final result.

Definite integration not only computes accumulation but also plays a crucial role in determining quantities like areas and solving physics problems related to motion.