Question

Evaluate the following definite integrals. $$\int_{0}^{\pi / 4}\left(\sec ^{2} t \mathbf{i}-2 \cos t \mathbf{j}-\mathbf{k}\right) d t$$

Short answer

Question: Evaluate the following definite integral: $$\int_{0}^{\pi/4}\left(\sec ^{2} t \mathbf{i}-2 \cos t \mathbf{j}-\mathbf{k}\right) d t$$ Answer: The definite integral evaluates to: $$\textbf{i} - \sqrt{2}\textbf{j} - \frac{\pi}{4}\textbf{k}.$$

Step-by-step solution

Component 1: \(\sec ^{2} t \mathbf{i}\)

To integrate the first component, we need to find the antiderivative of \(\sec^2{t}\) with respect to t. The antiderivative of \(\sec^2{t}\) is \(\tan{t}\). Now, we can evaluate the definite integral for the first component: $$\int_{0}^{\pi/4}\sec^2{t} \, dt = \left[\tan{t}\right]_0^{\pi/4} = \tan{\frac{\pi}{4}} - \tan{0} = 1-0 = 1$$ So, the first component is \(\textbf{i}\).

Component 2: \(-2\cos{t}\mathbf{j}\)

To integrate the second component, we need to find the antiderivative of \(-2\cos{t}\) with respect to t. The antiderivative of \(-2\cos{t}\) is \(-2\sin{t}\). Now, we can evaluate the definite integral for the second component: $$\int_{0}^{\pi/4}(-2\cos{t})\, dt = \left[-2\sin{t}\right]_0^{\pi/4} = -2\sin{\frac{\pi}{4}} - (-2\sin{0}) = -2\left(\frac{\sqrt{2}}{2}\right) = -\sqrt{2}$$ So, the second component is \(-\sqrt{2}\textbf{j}\).

Component 3: \(\mathbf{k}\)

The third component is already given as a constant. We need to find the antiderivative of \(-1\) with respect to t. The antiderivative is \(-t\). Now, we can evaluate the definite integral for the third component: $$\int_{0}^{\pi/4}(-1)\, dt = \left[-t\right]_0^{\pi/4} = -\frac{\pi}{4}$$ So, the third component is \(-\frac{\pi}{4}\textbf{k}\). Finally, we can combine the components to get the result of the definite integral: $$\int_{0}^{\pi/4}\left(\sec ^{2} t \mathbf{i}-2 \cos t \mathbf{j}-\mathbf{k}\right) d t = \textbf{i} - \sqrt{2}\textbf{j} - \frac{\pi}{4}\textbf{k}.$$

Key concepts

Integration Techniques

Understanding various integration techniques is essential for evaluating definite integrals, as different functions require different approaches. In the given exercise, we encounter a vector-valued function with components that involve trigonometric functions and constants.

For the first component, we use a direct integral that leverages the basic antiderivative of \(\sec^2{t}\), which is \(\tan{t}\). In contrast, the second component requires a basic understanding of integrating trigonometric functions, specifically integrating \(\cos{t}\) to obtain \(\sin{t}\). The third component is a constant, and integrating a constant is straightforward: the antiderivative of \(1\) is \(t\), making the component ease to integrate over the provided limits.

For a student to fully grasp these techniques, practice with various types of integrals, such as polynomial, exponential, and inverse trigonometric functions, is advised. This reinforcement helps solidify the recognition and application of appropriate integration methods.

Antiderivatives

The concept of an antiderivative plays a vital role in solving definite integrals. An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function you started with. In this exercise, knowing that \(\sec^2{t}\) is the derivative of \(\tan{t}\) and \(\cos{t}\) is the derivative of \(\sin{t}\) enables us to find the antiderivatives required for the solution.

When we solve for definite integrals, we are essentially finding the difference between the antiderivative evaluated at the upper and lower bounds of integration. This is known as the Fundamental Theorem of Calculus. Thus, for \(\sec^2{t}\) the antiderivative is calculated over the interval \( [0, \pi/4] \) to yield the first component's definite integral.

Trigonometric Integrals

Solving definite integrals that involve trigonometric functions, known as trigonometric integrals, can sometimes be tricky. However, the trigonometric integrals in our exercise involve basic trigonometric functions, which are easier to handle.

For example, the integral of \(\sec^2{t}\) is straightforward, as it is a common integral that yields \(\tan{t}\). Meanwhile, the integral of \(\cos{t}\), another fundamental trigonometric integral, gives us \(\sin{t}\). Learning the basic trigonometric integrals, like those of \(\sin{t}\), \(\cos{t}\), and \(\tan{t}\), is particularly useful as they often serve as the building blocks for more complex integrals involving trigonometric identities.

To reinforce these concepts, create flashcards of fundamental antiderivatives or regularly practice by integrating various trigonometric functions. This will help in recognizing patterns and applying the correct techniques in future integration problems.