Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the integral. \(\int \cos t+\sec t \tan t d t\)

Step-by-Step Solution

Answer

The value of the integral is \(\sin(t) + \sec(t) + C\).

Step 1: Compute Antiderivative

Remembering that the antiderivative of \(\cos(t)\) is \(\sin(t)\), the antiderivative of the first term is just \(\sin(t)\). Observing that the second term, \(\sec(t)\tan(t)\), is the derivative of \(\sec(t)\), this means that its antiderivative is simply \(\sec(t)\). So the antiderivative of the original fraction is \(\sin(t) + \sec(t)\).

Step 2: Suite Integral

Applying the Fundamental Theorem of Calculus, we note that any definite integral of a function f from a to b could be found by calculating the antiderivative at b, subtracting the antiderivative at a. Here, since we are dealing with an indefinite integral, there is no need to evaluate at certain limits. Therefore, the integral simplifies to be \(\sin(t) + \sec(t) + C\), where C is the constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematical analysis that focuses on accumulation and the process of adding up small parts to find the whole. It is particularly concerned with finding the areas under curves, volumes of shapes, and solutions to differential equations. When you encounter an integral, it asks you to aggregate tiny slices of a function's output over a specific range, or possibly over all values.

• Definite Integrals: Calculate exact values over a specified interval. For example, the area under a curve from point \(a\) to \(b\) on the x-axis.
• Indefinite Integrals: Represent a family of functions, including an arbitrary constant \(C\), indicating all possible antiderivatives of a function.

In the provided exercise, the integral \(\int \cos t + \sec t \tan t \, dt\) is an indefinite integral, as it seeks to find the general form of the antiderivative without specified limits.

Antiderivative

An antiderivative of a function is simply another function whose derivative is the original function. The process of finding the antiderivative is also called integration. When you integrate a function, you reverse the process of differentiation. For instance, since the derivative of \(\sin(t)\) is \(\cos(t)\), the antiderivative of \(\cos(t)\) is \(\sin(t)\).

• In the given exercise, the antiderivative of \(\cos(t)\) is \(\sin(t)\).
• The antiderivative of \(\sec(t)\tan(t)\), recognizing it as the derivative of \(\sec(t)\), is \(\sec(t)\).

Therefore, the antiderivative of the entire expression \(\int \cos t + \sec t \tan t \, dt\) is \(\sin(t) + \sec(t) + C\), where \(C\) is a constant. This process is a critical part of solving integrals, especially in the context of indefinite integrals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, two main aspects of calculus. It states that differentiation and integration are essentially inverse processes. This theorem is powerful because it provides a way to evaluate definite integrals as the difference of the values of an antiderivative at two points.
This theorem has two parts:

• First Part: If a function is continuous over an interval, the integral of that function over the interval can be found using an antiderivative.
• Second Part: The derivative of an integral function is the original function itself.

In the provided problem, we deal with an indefinite integral, meaning we don't evaluate over specific limits. Here, the role of the Fundamental Theorem is in recognizing the relationship between finding antiderivatives and interpreting integrals.