Problem 35 - Advanced Engineering Mathematics

Question

In Problems, evaluate the given integral. $$ \int\left(t e^{t} \mathbf{i}-e^{-2 t} \mathbf{j}+t e^{t^{2}} \mathbf{k}\right) d t $$

Step-by-Step Solution

Answer

The integral is \( (t e^{t} - e^{t}) \mathbf{i} - \frac{1}{2} e^{-2t} \mathbf{j} + \frac{1}{2} e^{t^2} \mathbf{k} + \mathbf{C} \).

Step 1: Break Down the Integral

The integral can be split into three separate integrals for each component of the vector function. This means you can handle each part independently. 1. For the \( \mathbf{i} \) component: \( \int t e^{t} \, dt \)2. For the \( \mathbf{j} \) component: \( - \int e^{-2t} \, dt \)3. For the \( \mathbf{k} \) component: \( \int t e^{t^2} \, dt \)

Step 2: Evaluate the \( \mathbf{i} \) Component

Use integration by parts for \( \int t e^{t} \, dt \). Let \( u = t \) and \( dv = e^{t} \, dt \). Then \( du = dt \) and \( v = e^{t} \).The formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \]Apply the formula:\[ \int t e^{t} \, dt = t e^{t} - \int e^{t} \, dt = t e^{t} - e^{t} + C_1 \] where \( C_1 \) is the integration constant.

Step 3: Evaluate the \( \mathbf{j} \) Component

The integral \( - \int e^{-2t} \, dt \) can be solved using the substitution method. Let \( u = -2t \), then \( du = -2 \, dt \) or \( dt = \frac{-1}{2} du \).Substitute and solve:\[ - \int e^{-2t} \, dt = \frac{-1}{2} \int e^{u} \, du = \frac{-1}{2} e^{u} + C_2 \]Substitute back for \( u \):\[ = \frac{-1}{2} e^{-2t} + C_2 \] where \( C_2 \) is the integration constant.

Step 4: Evaluate the \( \mathbf{k} \) Component

The integral \( \int t e^{t^2} \, dt \) can be solved using substitution. Let \( u = t^2 \), then \( du = 2t \, dt \) or \( t \, dt = \frac{1}{2} \, du \).Substitute and solve:\[ \int t e^{t^2} \, dt = \frac{1}{2} \int e^{u} \, du = \frac{1}{2} e^{u} + C_3 \]Substitute back for \( u \):\[ = \frac{1}{2} e^{t^2} + C_3 \] where \( C_3 \) is the integration constant.

Step 5: Write the Final Integral Result

Combine all the components back into vector form:\[ \left( t e^{t} - e^{t} + C_1 \right) \mathbf{i} - \left( \frac{1}{2} e^{-2t} + C_2 \right) \mathbf{j} + \left( \frac{1}{2} e^{t^2} + C_3 \right) \mathbf{k} \]These constants \( C_1, C_2, \) and \( C_3 \) can be combined into a single constant vector. So, the final answer is:\[ \left( t e^{t} - e^{t} \right) \mathbf{i} - \frac{1}{2} e^{-2t} \mathbf{j} + \frac{1}{2} e^{t^2} \mathbf{k} + \mathbf{C} \], where \( \mathbf{C} \) is a constant vector.

Key Concepts

Integration by Parts

Integration by parts is a technique used to solve integrals where the standard method of integration does not apply directly. It is particularly useful when you encounter the product of two different types of functions, such as algebraic and exponential or logarithmic. The essence of the formula is derived from the product rule for differentiation and is given by: \[\int u \, dv = uv - \int v \, du\]Here's how to apply it:

• Select \( u \) and \( dv \) from the integral \( \int u \, dv \).
• Differentiate to find \( du \), and integrate \( dv \) to get \( v \).
• Substitute these into the formula, solve the resulting integral, and simplify if necessary.

For example, in the exercise, to solve \( \int t e^{t} \, dt \), we choose \( u = t \) and \( dv = e^{t} \, dt \). This simplifies the integral and allows us to find a solution efficiently.

Substitution Method

The substitution method is a fundamental technique in integration, often used to simplify integrals by replacing a complicated part of the integral with a single variable. This method is particularly effective for integrals that resemble the result of the chain rule. To use the substitution method, follow these steps:

• Select a new variable, \( u \), to replace a more complex expression.
• Differentiate \( u \) to find \( du \), and express \( dt \) (or the differential component) in terms of \( du \).
• Substitute \( u \) and \( dt \) with their new expressions in the integral.
• Integrate with respect to \( u \), then substitute back to the original variable.

Consider the integral \( - \int e^{-2t} \, dt \). By letting \( u = -2t \), you can convert and simplify the expression, making it more manageable to integrate, with the solution ultimately requiring substitution back to the original variable.

Vector Integrals

Vector integrals involve integrating vector functions, where each component of the vector is an independent scalar function that needs to be integrated separately. These types of integrals are common in physics and engineering, particularly when dealing with fields or forces. To evaluate vector integrals, dissect the vector into its components:

• Separate the integral for each vector component.
• Evaluate each scalar integral independently using appropriate techniques.
• Recombine the results into a vector form.

In our exercise, the vector \( t e^{t} \mathbf{i} - e^{-2t} \mathbf{j} + t e^{t^{2}} \mathbf{k} \) is broken into three integrals corresponding to \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) components. Each component is dealt with using specific integration techniques, and then the solutions are combined back into the vector format.

Integration Techniques

Mastering integration techniques is crucial, as it enables you to tackle a wide range of functions and applications. Key techniques include:

• Integration by Parts: Best used for integrals of products of functions, letting you divide and conquer different parts of the integral.
• Substitution Method: Allows simplification by swapping complex expressions with simple variables.
• Trigonometric Integrals: Useful for solving integrals involving sine, cosine, and other trigonometric functions.
• Partial Fraction Decomposition: Effective for rational functions, decomposing them into simpler fractions that are easy to integrate.

To choose the right technique, consider the form and complexity of the integrand. In our problem, integration by parts and substitution methods were employed for different components, demonstrating the strategic selection of techniques to simplify and solve the given vector integral.