Question

For the following functions \(f\), find the anti-derivative \(F\) that satisfies the given condition. $$f(t)=\sec ^{2} t ; F(\pi / 4)=1,-\pi / 2 < t < \pi / 2$$

Short answer

Question: Find the anti-derivative \(F(t)\) of the function \(f(t) = \sec^2{t}\) that satisfies the condition \(F(\pi/4) = 1\). Answer: The anti-derivative \(F(t)\) of the given function is \(F(t) = \tan{t}\).

Step-by-step solution

Find the general anti-derivative of \(f(t)\)

Recall that the derivative of the tangent function is $$\frac{d}{dt}(\tan{t}) = \sec^{2}{t}.$$ So, the anti-derivative of \(\sec^2{t}\) is \(\tan{t}\) plus a constant of integration. Thus, we can write the general anti-derivative as $$F(t) = \tan{t} + C.$$

Use the given condition to find the constant of integration

We are given the condition that \(F(\pi/4) = 1\). So, we can plug in \(\pi/4\) for \(t\) in our expression for \(F(t)\) and solve for \(C\): $$1 = \tan{(\pi/4)} + C.$$ Since \(\tan{(\pi/4)} = 1\), we have $$1 = 1 + C,$$ which implies that \(C = 0\).

Write the final anti-derivative

Now that we have found the constant of integration, we can write the final anti-derivative as $$F(t) = \tan{t}.$$

Key concepts

Integral Calculus

Integral calculus is a fundamental branch of mathematics, which focuses on finding the anti-derivatives of functions. An anti-derivative, also known as an indefinite integral, of a function is essentially the reverse of taking a derivative. Let's delve into what this means with an example.

Consider we have a function represented as a curve on a graph, and we wish to understand the accumulation of the area under that curve. Integral calculus is the tool that allows us to calculate that area. In the process of integration, we come across an infinite array of possible functions that could represent this area since a constant can be added without affecting the derivative. Therefore, when we integrate a function, we have to include a 'constant of integration' to represent all these possibilities.

Integrating the function in the given exercise, we found that the anti-derivative of the secant squared function, denoted as \(\sec^2 t\), is the tangent function \(\tan t\). However, because any constant added to this result has a derivative of zero, the integral is not complete without accounting for this constant, thus we write the anti-derivative as \(\tan t + C\), where \(C\) is the constant of integration.

Secant Function

The secant function, denoted as \(\sec x\), is one of the trigonometric functions that can be thought of as the reciprocal of the cosine function, meaning \(\sec x = \frac{1}{\cos x}\). In the context of integral calculus, certain trigonometric functions have standard anti-derivatives. For instance, as shown in our example, the anti-derivative of the secant squared function \(\sec^2 x\) is well-known among students and mathematicians alike to be the tangent function \(\tan x\).

Understanding the behavior of the secant function is important as it exhibits different characteristics than other trigonometric functions. For example, its graph does not resemble the familiar sine or cosine wave, and it has vertical asymptotes where the cosine function is zero. These properties must be considered when integrating expressions involving the secant function, as the limits of integration must avoid these asymptotes to ensure that the integral is properly defined.

Constant of Integration

The constant of integration is a fundamental component of the indefinite integral. Whenever we find the anti-derivative of a function, we do not obtain a single function but rather a family of functions that are vertically shifted from each other by some constant amount. This is expressed mathematically as adding an arbitrary constant \(C\) to the anti-derivative.

In our exercise, after integrating the secant squared function \(\sec^2 t\), we must include this constant of integration in our general solution, which is initially \(F(t) = \tan t + C\). To pinpoint the exact member of our family of functions that satisfies a specific condition, such as \(F(\pi/4) = 1\), we solve for \(C\) using the given condition. In this case, the value of \(C\) happened to be zero, leading us to our particular solution: \(F(t) = \tan t\). The constant of integration is thus what allows us to cater our general anti-derivative to specific scenarios or initial conditions.