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Chapter 28: Problem 3

Integrate each of the given functions. $$\int \cos 2 x d x$$

Short Answer

Expert verified
\( \frac{1}{2} \sin(2x) + C \)

Step by step solution

01

Identify the Integral Formula for Cosine

The integral of \(\int \cos(ax)\, dx\) is a standard integral formula, which evaluates to \(\frac{1}{a}\sin(ax) + C\), where \(a\) is a constant and \(C\) is the constant of integration.
02

Apply the Constant Multiple Rule

In the given integral, \(a = 2\), so according to the formula, we can factor out \(\frac{1}{2}\) from the integral:\[\int \cos(2x)\, dx = \frac{1}{2} \int \cos(2x)\, dx\]
03

Integrate the Function

Now, recognize that the inner integral is simply \(\sin(2x)\), by the integral formula for cosine:\[\frac{1}{2} \int \cos(2x)\, dx = \frac{1}{2} \cdot \sin(2x)\]
04

Add the Constant of Integration

Finally, append the constant of integration \(C\) to account for the family of antiderivatives:\[\frac{1}{2} \sin(2x) + C\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function Integration
When we talk about integrating a cosine function, specifically the cosine of a multiple of a variable, we are dealing with a well-known formula. The formula we'll apply here is the integral of \( \int \cos(ax) \, dx \), where \( a \) is a constant. This integral evaluates to \( \frac{1}{a} \sin(ax) + C \). The meaning of this is straightforward:
  • First, identify the coefficient \( a \) of \( x \) in the cosine function.
  • Then, convert the cosine integral into a sine function.
  • Multiply the result by \( \frac{1}{a} \) to account for the coefficient.
In our given example, since we have \( \cos(2x) \), we recognize that \( a = 2 \). Consequently, this integral process turns \( \cos(ax) \) into \( \sin(ax) \), and we end up with \( \frac{1}{2} \sin(2x) \). This conversion hinges on the relationship between the derivative of sine and the integral of cosine.
Antiderivative
The term "antiderivative" is crucial in understanding integrals. Simply put, an antiderivative of a function is another function whose derivative returns the original function. In the context of our cosine example, we are looking for a function whose derivative is \( \cos(2x) \).
  • Start by identifying the integrand, which is \( \cos(2x) \).
  • Using the antiderivative formula \( \int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C \), we're essentially "reversing" differentiation.
  • The result is \( \frac{1}{2} \sin(2x) \), which implies \( \frac{1}{2} \sin(2x) \) is an antiderivative of \( \cos(2x) \).
Antiderivatives can be a bit perplexing at first, because there isn't just one. Rather, they belong to a **family of functions** that differ by a constant, since derivatives of constants are zero. This leads us to the next important concept: the constant of integration.
Constant of Integration
The constant of integration, represented by \( C \), appears in the result of an indefinite integral. It's crucial to include \( C \) because integration, as the inverse of differentiation, involves a family of functions. Each member differs by a constant, which is invisible when taking derivatives.Here are some key points to consider:
  • The antiderivative of a function is not unique; any constant added to an antiderivative is also an antiderivative.
  • \( C \) represents the infinite set of constants that can be added to satisfy different initial conditions or to adjust a solution.
  • When graphing, this family manifests as vertical shifts of the curve represented by the antiderivative.
In our example, the constant \( C \) is added to the result \( \frac{1}{2} \sin(2x) \) to indicate that this function is part of a larger set of solutions. This ensures completeness and accuracy of answers when solving integration problems.

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Most popular questions from this chapter

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