Question

Evaluate the given definite or indefinite integral. $$ \int\left\langle t^{3}, \cos t, t e^{t}\right\rangle d t $$

Short answer

\(\int \langle t^3, \cos t, t e^t \rangle \, dt = \langle \frac{t^4}{4}, \sin t, t e^t - e^t \rangle + \langle C_1, C_2, C_3 \rangle\)

Step-by-step solution

Understand the Problem

We are given a vector function with three components: \(t^3\), \(\cos t\), and \(t e^t\). We need to evaluate the indefinite integral of this vector-valued function.

Integrate Each Component Separately

The vector integral \(\int \langle t^3, \cos t, t e^t \rangle \, dt\) can be calculated by integrating each component separately. We will have three separate integrals to solve:1. \(\int t^3 \, dt\)2. \(\int \cos t \, dt\)3. \(\int t e^t \, dt\)

Solve \(\int t^3 \, dt\)

To find the integral of \(t^3\), we use the power rule: \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\). So:\[\int t^3 \, dt = \frac{t^{4}}{4} + C_1 \]

Solve \(\int \cos t \, dt\)

The integral of \(\cos t\) is a standard trigonometric integral. Using the formula \(\int \cos t \, dt = \sin t + C\), we have:\[\int \cos t \, dt = \sin t + C_2\]

Solve \(\int t e^t \, dt\)

The integral of \(t e^t\) needs integration by parts, where \(u = t\) and \(dv = e^t \, dt\). This gives:- \(du = dt\)- \(v = e^t\)Using integration by parts formula \(\int u \, dv = uv - \int v \, du\), we have:\[\int t e^t \, dt = t e^t - \int e^t \, dt = t e^t - e^t + C_3 \]

Combine the Results

The definite integral of the vector function \(\int \langle t^3, \cos t, t e^t \rangle \, dt\) combines the integrals of the individual components:\[\int \langle t^3, \cos t, t e^t \rangle \, dt = \left\langle \frac{t^4}{4} + C_1, \sin t + C_2, t e^t - e^t + C_3 \right\rangle\]

Key concepts

Vector-Valued Functions

Vector-valued functions play a significant role in vector calculus. They are functions that have vectors as their output, rather than scalar values. A typical vector-valued function is expressed as \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\), where \(x(t), y(t),\) and \(z(t)\) are scalar functions of the variable \(t\). Each of these scalar functions represents a component of the vector function.

• Vector-valued functions can describe a path or trajectory in space, making them useful in physics and engineering.
• They allow the representation of more complex entities as they encapsulate multi-dimensional data in a single mathematical expression.

In our example, we have the vector function \(\langle t^3, \cos t, t e^t \rangle\), which implies a curve or path in a three-dimensional space. Understanding each component will help you evaluate the vector as a whole.

Integration by Parts

Integration by parts is a useful technique in calculus, especially when dealing with products of functions. This method is derived from the product rule for differentiation and is expressed by the formula:\[ \int u \, dv = uv - \int v \, du \]Here, you choose one part of the product to differentiate (\(u\)) and the other to integrate (\(dv\)).

In our given solution, integration by parts is applied to the integral \(\int t e^t \, dt\). In this case:

• Choose \(u = t\), which when differentiated gives \(du = dt\).
• Choose \(dv = e^t \, dt\), which when integrated gives \(v = e^t\).

Applying the integration by parts results in:\[ \int t e^t \, dt = t e^t - \int e^t \, dt = t e^t - e^t + C \]This process simplifies the integration of the product of functions, turning an otherwise complex integral into a manageable one.

Indefinite Integrals

Indefinite integrals involve finding the antiderivative of a function. It represents a family of functions, differing only by a constant \(C\). The purpose is to reverse the differentiation process.

When evaluating indefinite integrals, such as \(\int f(x) \, dx\), it is crucial to determine a function whose derivative is equal to the integrand \(f(x)\). The result is written as \(F(x) + C\), where \(F(x)\) is the antiderivative.

In our example, we solve the indefinite integrals of vector components individually:

• For \(\int t^3 \, dt\), apply the power rule: \(\frac{t^4}{4} + C_1\).
• For \(\int \cos t \, dt\), use known results: \(\sin t + C_2\).
• For \(\int t e^t \, dt\), utilize integration by parts: \(t e^t - e^t + C_3\).

Each component's result forms part of the final vector integral. The constants \(C_1, C_2,\) and \(C_3\) represent the general solutions from each integral, indicating that any antiderivative differs by only a constant.