Problem 22 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int e^{-3 t} \sin 4 t d t\)

Step-by-Step Solution

Answer

\( \int e^{-3t} \sin 4t \, dt = \frac{e^{-3t}}{25} (-3 \sin(4t) - 4 \cos(4t)) + C \).

Step 1: Recognize the Integral Type

Notice that the integral \( \int e^{-3t} \sin 4t \, dt \) is of the form \( \int e^{at} \sin(bt) \, dt \). This matches a common form found in most integral tables.

Step 2: Identify the Appropriate Formula

In the integral table, find the formula for \( \int e^{ax} \sin(bx) \, dx \), usually given as: \[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C \]. Here, \( a = -3 \) and \( b = 4 \).

Step 3: Substitute the Appropriate Constants

Plug \( a = -3 \) and \( b = 4 \) into the formula: \[ \frac{e^{-3t}}{(-3)^2 + 4^2} ((-3) \sin(4t) - 4 \cos(4t)) + C \].

Step 4: Simplify the Expression

Calculate \( (-3)^2 + 4^2 = 9 + 16 = 25 \) and simplify the expression: \[ \frac{e^{-3t}}{25} (-3 \sin(4t) - 4 \cos(4t)) + C \].

Step 5: Write the Final Result

Therefore, the evaluated integral is: \[ \int e^{-3t} \sin 4t \, dt = \frac{e^{-3t}}{25} (-3 \sin(4t) - 4 \cos(4t)) + C \].

Key Concepts

Integration by Formula

In integral calculus, one efficient method for finding integrals is the use of integration formulas. Integration by formula simplifies the integration process by using established formulas for specific types of integrals. This is especially helpful where direct integration may involve complex steps or advanced techniques. When faced with an integral, the first step is to identify it in a form resembling one of these known formulas.

• Integration tables list standard forms, allowing you to substitute variables and constants directly to obtain the integral.
• These forms are derived from fundamental integration rules, which make them trustworthy for obtaining accurate results swiftly.

Integration by formula thus acts as a shortcut, but it requires practice to recognize patterns and appropriate applications. In the provided exercise, recognizing the integral as matching a standard form simplifies the problem significantly.

Exponential Functions

Exponential functions play a crucial role in various fields of mathematics and science. In general, exponential functions have the form \(e^{kx}\), where \(e\) is Euler's number (approximately 2.71828), and \(k\) is a constant. They exhibit continuous growth or decay, depending on the value of \(k\).

• In integrals, exponential functions often appear when modeling phenomena such as radioactive decay, population growth, and certain economic models.
• The derivative and the integral of \(e^{kx}\) maintain similar forms, making them quite manageable in calculus.

In our exercise, we deal with the function \(e^{-3t}\), highlighting exponential decay. When integrating exponential functions in conjunction with other functions, such as sines and cosines, formula-based integration can greatly assist in simplifying the process.

Trigonometric Integrals

Trigonometric integrals involve integrating expressions containing trigonometric functions like sine, cosine, tangent, etc. These integrals are crucial when dealing with periodic functions and oscillatory behavior in physics and engineering.

Common trigonometric functions include \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), and others, often forming part of more complex integrals.

Understanding the properties, such as periodicity and symmetry, of these functions can be very useful when solving integrals.

For the exercise in question, we used the sine function, which represents an oscillating behavior. Trigonometric integrals often appear together with exponential functions, as seen in our example, yielding integrals commonly found in signal processing and other oscillatory phenomena. Recognizing these combined forms is key, making the use of integral tables highly beneficial for quick and accurate solutions.