Problem 46 - Thomas Calculus

Question

In Exercises \(17-56,\) find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(2 \cos 2 x-3 \sin 3 x) d x$$

Step-by-Step Solution

Answer

The general antiderivative is \( \sin 2x - \cos 3x + C \).

Step 1: Identify the integral

The given expression to integrate is \( \int (2 \cos 2x - 3 \sin 3x) \, dx \).

Step 2: Split the integral

Use the property of linearity of integrals to split the expression into two separate integrals: \[\int 2 \cos 2x \, dx - \int 3 \sin 3x \, dx.\]

Step 3: Integrate each term separately

First, consider \( \int 2 \cos 2x \, dx \): The antiderivative of \( \cos ax \) is \( \frac{1}{a} \sin ax \). Therefore, the antiderivative is \( 2 \cdot \frac{1}{2} \sin 2x = \sin 2x + C_1\). Now integrate \( \int 3 \sin 3x \, dx \): The antiderivative of \( \sin ax \) is \( -\frac{1}{a} \cos ax \). Therefore, the antiderivative is \( 3 \cdot -\frac{1}{3} \cos 3x = -\cos 3x + C_2\).

Step 4: Combine antiderivatives and introduce constant of integration

Combine the antiderivatives to get the general antiderivative of the original integral: \[\int (2 \cos 2x - 3 \sin 3x) \, dx = \sin 2x - \cos 3x + C,\] where \( C = C_1 - C_2 \) is the constant of integration.

Step 5: Verify by differentiation

Differentiate \( f(x) = \sin 2x - \cos 3x + C \): Recall the derivative rules: \( (\sin ax)' = a \cos ax \) and \( (-\cos ax)' = a \sin ax \). \( f'(x) = (2 \cos 2x + 3 \sin 3x) \), which matches the original integrand \( 2 \cos 2x - 3 \sin 3x \).

Key Concepts

Antiderivative

An antiderivative is essentially the reverse process of taking a derivative. When you have a function and perform its derivative, you get a new function. Finding the antiderivative allows you to go back to the original function from its derivative. It is important to realize that antiderivatives are not unique. A function can have many antiderivatives that differ only by a constant.Consider the simplest example, the function sin(x). Its derivative is cos(x). So, any function that, when differentiated, gives you cos(x) is an antiderivative of cos(x). This means that the function sin(x) + C is an antiderivative of cos(x), where C is a constant. Antiderivatives are closely linked to indefinite integrals. When you find the indefinite integral of a function, you are essentially looking for its antiderivative. The notation for finding an antiderivative is the integral symbol with no limits, such as \( \int f(x) \, dx \). This operation will yield the antiderivative plus a constant of integration.

Linearity of Integrals

The linearity of integrals is a useful property that allows you to easily break down and solve complex integrals by handling each term separately. This property states that the integral of a sum is equal to the sum of the integrals of the individual terms.For example, when you have an integral such as \( \int (a f(x) + b g(x)) \, dx \), you can split it into \( a \int f(x) \, dx + b \int g(x) \, dx \). This makes integration much simpler because you don't have to handle the entire expression at once.In this exercise, the expression \( \int (2 \cos 2x - 3 \sin 3x) \, dx \) is split into two separate integrals: \( \int 2 \cos 2x \, dx - \int 3 \sin 3x \, dx \). By doing this, you can work on smaller, more manageable pieces. This step-by-step application of linearity simplifies the overall process of finding the antiderivative.

Trigonometric Integration

Trigonometric integration deals with finding the integrals of trigonometric functions. Mastering this process is helpful because trigonometric functions frequently appear in calculus.The antiderivatives of basic trigonometric functions are essential building blocks for more complex integrals:

• The antiderivative of \( \cos(ax) \) is \( \frac{1}{a}\sin(ax) \).
• The antiderivative of \( \sin(ax) \) is \( -\frac{1}{a}\cos(ax) \).

In this exercise, when integrating \( \int 2 \cos 2x \, dx \), the antiderivative is computed as \( \sin 2x \) because the coefficient comes out to cancel the 2 in the derivative. Similarly, for the term \( -3 \sin 3x \), the antiderivative is \( -\cos 3x \). These integrals demonstrate the elegant transformations that can occur when integrating trigonometric functions.

Constant of Integration

When you find an indefinite integral, it’s crucial to include an arbitrary constant, often denoted as \( C \). This constant represents all possible vertical shifts in the family of antiderivative functions. It arises because differentiation nullifies any constant; hence, all functions differing by only a constant have the same derivative.Each time you compute an indefinite integral, the constant is added at the end of your expression. In this exercise's final solution, \( \int (2 \cos 2x - 3 \sin 3x) \, dx = \sin 2x - \cos 3x + C \), the \( C \) signifies the constant of integration.Not including the constant can lead to incorrect interpretations or solutions related to initial conditions, especially in differential equations and problems with specific constraints. Always remember to add the constant when dealing with indefinite integrals."