Question

How do you find the indefinite integral of \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?\)

Short answer

Question: Find the indefinite integral of the vector-valued function \(\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle\). Answer: The indefinite integral of the vector-valued function \(\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle\) is \(\int{\mathbf{r}(t) \, dt} = \langle F(t) + C_1, G(t) + C_2, H(t) + C_3 \rangle\), where F(t), G(t), and H(t) are the antiderivatives of f(t), g(t), and h(t), respectively, and C_1, C_2, and C_3 are the constants of integration.

Step-by-step solution

Identify the component functions

The given vector-valued function is \(\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle\). The component functions f(t), g(t), and h(t) each represent a scalar function of the parameter t.

Integrate each component function

To find the indefinite integral of the vector-valued function, we need to find the indefinite integral of each component function. Integrate f(t), g(t), and h(t) with respect to t separately, as follows: \[ \int{f(t) \, dt} = F(t) + C_1 \\ \int{g(t) \, dt} = G(t) + C_2 \\ \int{h(t) \, dt} = H(t) + C_3 \] where F(t), G(t), and H(t) are the antiderivatives of f(t), g(t), and h(t), respectively, and C_1, C_2, and C_3 are the constants of integration.

Construct the indefinite integral of the vector-valued function

Combine the antiderivatives F(t), G(t), and H(t) with their respective constants of integration to form the indefinite integral of the vector-valued function: \[ \int{\mathbf{r}(t) \, dt} = \langle F(t) + C_1, G(t) + C_2, H(t) + C_3 \rangle \] This is the indefinite integral of the given vector-valued function.

Key concepts

Integration of Functions

Integration is a fundamental concept in calculus, closely related to the idea of finding areas under curves and the accumulation of quantities. To understand the process of integration, consider a scalar function, say, f(t), which describes some quantity with respect to an independent variable, t. The indefinite integral, often represented by \[ \int f(t) \, dt \], essentially tells us about the function whose rate of change (derivative) is f(t).

It's like reconstructing the total distance traveled when you are given the speed over time but not the exact path taken. Since the process does not specify the starting point, the result includes a constant of integration, C, representing the unknown initial condition. In practice, one would apply integration techniques like substitution or partial fractions to find the antiderivative, which is a function whose derivative is the given function.

The step-by-step solution makes use of this foundational knowledge, where each component function, f(t), g(t), and h(t), is integrated separately to find the antiderivatives. These antiderivatives are then used to build up the indefinite integral of the vector-valued function.

Antiderivatives

An antiderivative, also known as an indefinite integral, reflects the 'reverse process' of differentiation. When you differentiate a function, you obtain the function's rate of change at any point. Antidifferentiation, conversely, aims to determine the original function given its rate of change. For instance, if \[ \frac{dF(t)}{dt} = f(t) \], then F(t) is an antiderivative of f(t).

It is essential to understand that antiderivatives are not unique; adding a constant to one antiderivative creates another. This is why we often see + C in the answer to an indefinite integral, signifying that family of functions that share the same derivative. The exercise demonstrates this principle by incorporating constants C_1, C_2, and C_3 for each independent antiderivative. These constants highlight the 'plus a constant' characteristic inherent to antiderivatives, which is crucial for students to recognize and apply.

Vector Calculus

Vector calculus extends the principles of calculus to vector-valued functions. A vector-valued function, like \[\mathbf{r}(t)\] in the exercise, maps a real number t to a vector in space. These functions come in handy in physics and engineering when dealing with quantities that have both magnitude and direction, such as velocity, force, or acceleration.

When you integrate a vector-valued function, you deal with each of its scalar component functions independently, as each component represents a dimension in space. The integral of the function equals the vector whose components are the integrals of the component functions, which captures the essence of vector calculus—treating multidimensional problems component-wise.

In the given solution, we integrate the components of \[\mathbf{r}(t)\] to find \[\int{\mathbf{r}(t) \, dt}\]. Each scalar function contributes to the vector's overall behavior, and through integration, we combine those contributions to understand the vector function's overall effect. This principle is not only crucial for solving mathematical problems but also for applications in multiple fields such as electromagnetism, fluid dynamics, and more.