Question

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle t^{4}-3 t, 2 t-1,10\right\rangle$$

Short answer

Answer: The indefinite integral of the given vector-valued function is \(\mathbf{R}(t) = \left\langle \frac{t^5}{5} - \frac{3t^2}{2} + C_1, t^2 - t + C_2, 10t + C_3 \right\rangle\), where \(C_1\), \(C_2\), and \(C_3\) are constants of integration.

Step-by-step solution

Separate the components of the vector

We can represent the given vector function, \(\mathbf{r}(t)\), as three separate scalar functions: $$f(t) = t^4 - 3t,$$ $$g(t) = 2t - 1,$$ $$h(t) = 10.$$

Integrate each component

Now, we will find the indefinite integral of each scalar function, \(f(t)\), \(g(t)\), and \(h(t)\): $$F(t) = \int (t^4 - 3t) dt,$$ $$G(t) = \int (2t - 1) dt,$$ $$H(t) = \int 10 dt.$$

Find the antiderivative of each component

Now, we will find the antiderivative of each of these scalar functions: $$F(t) = \frac{t^5}{5} - \frac{3t^2}{2} + C_1,$$ $$G(t) = t^2 - t + C_2,$$ $$H(t) = 10t + C_3,$$ where \(C_1\), \(C_2\), and \(C_3\) are constants of integration.

Combine the components

Finally, we will combine the results of our calculations above to form the new vector-valued function, \(\mathbf{R}(t)\): $$\mathbf{R}(t) = \left\langle \frac{t^5}{5} - \frac{3t^2}{2} + C_1, t^2 - t + C_2, 10t + C_3 \right\rangle.$$ This is the indefinite integral of the given vector-valued function, \(\mathbf{r}(t)\).

Key concepts

Indefinite Integral

When solving problems in calculus, indefinite integrals play a crucial role. An indefinite integral is the reverse operation of differentiation. It helps us find a function whose derivative is the given function. This process is also known as integration without limits.
For real-valued functions, the indefinite integral takes the form:

• \( \int f(x) \, dx = F(x) + C \)

Here, \(F(x)\) is the antiderivative of \(f(x)\), and \(C\) represents the constant of integration. Indefinite integrals help us discover families of functions that vary by this constant. This is important because the process of differentiation loses information about constant terms in the function, which integration restores.
When dealing with vector-valued functions, we apply these integration concepts component-wise, meaning each dimension of the vector is integrated separately.

Antiderivative

The concept of an antiderivative is foundational to understanding indefinite integrals. The antiderivative works as the inverse of differentiation. It reconstructs a function from its derivative. Another way to describe it is that if \(F(x)\) is an antiderivative of \(f(x)\), then \(F'(x) = f(x)\).
Let's explore how to find one. Consider \(f(x)\) as your function, and look for some function \(F(x)\) that satisfies the following:

• \(\frac{d}{dx}[F(x)] = f(x)\)

Remember that there are infinitely many antiderivatives for any given function. All differ by a constant. Thus, the notation \(F(x) + C\) captures this infinite family, where \(C\) is any real number. If you differentiate \(F(x) + C\), you still get \(f(x)\), as the derivative of a constant is zero.
In vector-valued functions, an antiderivative individually handles each component of the vector. This keeps each part's distinct identity while assembling the general solution.

Vector-Valued Function

Vector-valued functions expand the idea of a regular function to include output in vector form. In essence, they map a single input value, like \(t\), into a multi-dimensional vector.
These functions are represented as:

• \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \)

Where \(f(t)\), \(g(t)\), and \(h(t)\) are the component functions.
Understanding vector-valued functions involves operating on their components separately. They are embedded in contexts like physics and engineering, where motions or forces have directions represented by vectors.
To compute an indefinite integral for such functions, integrate each of the component functions one by one. Once integrated, combine the results to form the final vector, which will express the integrated relation. This step-by-step component-focused approach makes it easier to manage multifaceted problems common in real-world applications.