91Ó°ÊÓ

An indefinite integral is a fundamental concept in calculus that represents the antiderivative of a function. When we find the indefinite integral of a function, we are essentially performing the operation opposite to differentiation. This process involves finding a function whose derivative produces the original function we began with.

In the context of vector calculus, as shown in the problem, when we have the derivative function \( \mathbf{r}'(t) = 4 e^{2t} \mathbf{i} + 3 e^{t} \mathbf{j} \), finding the indefinite integral means determining a vector function \( \mathbf{r}(t) \) whose derivative yields \( \mathbf{r}'(t) \). This involves calculating \( \int \left( 4 e^{2t} \mathbf{i} + 3 e^{t} \mathbf{j} \right) dt \).

Performing this operation separately:

• The integral of \( 4 e^{2t} \mathbf{i} \) is \( 2 e^{2t} \mathbf{i} \).
• The integral of \( 3 e^{t} \mathbf{j} \) is \( 3 e^{t} \mathbf{j} \).

Remember, when performing indefinite integration, there is always a constant of integration added to the result.

The constant of integration is an essential element of indefinite integrals. Every time you take the indefinite integral of a function, you must add a constant, usually noted as \( C \). This constant represents any constant number that could have been differentiated away in the original function, and since the derivative of a constant is zero, it appears when integrating.

• In the context of our problem, the indefinite integral of \( \mathbf{r}'(t) \) resulted in \( 2 e^{2t} \mathbf{i} + 3 e^{t} \mathbf{j} + \mathbf{C} \), where \( \mathbf{C} \) is the constant of integration.
• Given that this is vector calculus, \( \mathbf{C} \) itself is a vector, capturing the constant values that apply to each component of the vector \( \mathbf{r}(t) \).

To find the precise value of this vector constant, utilize the initial condition provided: \( \mathbf{r}(0) = 2 \mathbf{i} \), which helps specify the particular solution from amongst the infinite family of solutions represented by the general solution with \( + C \).
Inserting \( t = 0 \) into the equation helps in solving for \( \mathbf{C} \), leading us to zero in on a specific solution.

Vector functions are functions where each element is a vector, representing quantities that have both a magnitude and a direction. In physics and engineering, vector functions are typically used to describe the position, velocity, and acceleration of objects in space.

In the given problem, \( \mathbf{r}(t) \) is a vector function. It represents a position vector in the form \( x(t) \mathbf{i} + y(t) \mathbf{j} \).

• The x-component is influenced by \( e^{2t} \), resulting in its part of the function being \( 2 e^{2t} \mathbf{i} \).
• The y-component grows with \( e^{t} \), leading to its respective term \( 3 e^{t} \mathbf{j} \).

Vector functions are essential for mapping changes in multi-dimensional spaces and finding particular solutions involving dynamic systems. By determining the vector function over time, it is possible to understand how different components evolve and interact with each other over time.

In solving such problems, identifying and considering every component of the vector function is crucial. This ensures a complete understanding of the system it represents.