Problem 56 - Applied Calculus

Question

Find the indefinite integrals. $$ \int\left(2 e^{x}-8 \cos x\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( 2e^x - 8 \sin x + C \).

Step 1: Identify the Integral Components

The integral provided is \( \int (2e^x - 8 \cos x) \, dx \). This expression is a sum of two separate functions: \( 2e^x \) and \( -8 \cos x \). We will find the integrals of each component separately and combine them.

Step 2: Integrate the First Component

The first term is \( 2e^x \). Recall that the integral of \( e^x \) is \( e^x \). Thus, the integral of \( 2e^x \) is:\[\int 2e^x \, dx = 2e^x + C_1\]where \( C_1 \) is the constant of integration.

Step 3: Integrate the Second Component

The second term is \( -8 \cos x \). Recall that the integral of \( \cos x \) is \( \sin x \). Therefore, the integral of \( -8 \cos x \) is:\[\int -8 \cos x \, dx = -8 \sin x + C_2\]where \( C_2 \) is another constant of integration.

Step 4: Combine Both Integrals

Add together the integrals obtained from both components. This yields:\[\int (2e^x - 8 \cos x) \, dx = 2e^x - 8 \sin x + C\]Combine the constants \( C_1 \) and \( C_2 \) into a single integration constant \( C \).

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, and it allows us to find the antiderivative of a given function. In other words, it lets us deduce a function whose derivative is the given function. When tackling indefinite integrals, we generally encounter a variety of functions which may require different techniques for integration. These techniques simplify the process of finding the antiderivative and are crucial for solving complex problems.

• **Basic Rule Application**: Often, the simplest integrals are solved using basic rules such as the integral of a power function or a constant multiplier.
• **List of Common Functions**: As seen in the current exercise, knowing the standard integrals for common functions like exponential and trigonometric functions is useful.
• **Sum and Difference Rule**: When dealing with complex expressions, breaking them down into sums or differences of simpler expressions can be advantageous.
• **Constant Factor Rule**: This rule states that you can factor out constants from an integral, as shown by integrating 2 times the exponential function separately.

Breaking down an expression into its components and using these techniques accordingly allows for a more systematic approach, ensuring you don’t overlook details.

Exponential Functions

Exponential functions, characterized by the constant base raised to a variable exponent, are pervasive in both mathematics and various applications in science, particularly in modeling growth and decay.For the exponential function, the base is often the natural exponential number, denoted as \( e \). The function presents unique characteristics:

• **Derivative and Integral**: The remarkable property of the natural exponential function \( e^x \) is that its derivative and integral are the same: \[ \frac{d}{dx}(e^x) = e^x \quad \text{and} \quad \int e^x \, dx = e^x + C \]
• **Constant Multipliers**: When integrating functions like \( 2e^x \,\int 2e^x\ dx = 2\int e^x\ dx \), it simplifies the process by treating the constant multiplier separately.

Understanding this function's rules not only helps with integration but also facilitates comprehension of exponential growth models and natural logarithms. Incorporating such tools into calculus problems enables students to solve them reliably.

Trigonometric Functions

Trigonometric functions like sine (\( \sin x \)) and cosine (\( \cos x \)) are vital for solving problems involving periodic or oscillatory behaviors. They show up frequently in both pure and applied mathematics due to their relationship with circle geometry and wave phenomena.When dealing with these functions in calculus:

• **Integral of Cosine**: The integration of \( \cos x \) results in \( \sin x + C \), since it is the antiderivative of cosine.In the given exercise, we integrated \(-8 \\cos x \) \ and obtained \(-8 \sin x + C \).
  This often requires attention to constant multipliers.
• **Periodic Nature**: Understand that the trigonometric functions have periodic properties which can influence the integration when part of more complex expressions.

These concepts help create a solid foundation for calculating areas under curves involving trigonometric waves and help to solve physics problems involving harmonic motion.