Problem 35 Find the following indefinite in... [FREE SOLUTION]

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Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

$Math Studyset 91影视 Explanations$ Math

14 Edition

Chapter 8: Problem 35

Find the following indefinite integrals. $$\int \cos 2 x \, d x$$

Short Answer

Expert verified

$ \frac{1}{2} \text{sin}(2x) + C $

Step by step solution

Identify the integral form

The integral is of the form $\text{鈭珆 \text{cos}(kx) \, dx$, where $k$ is a constant.

Recall the antiderivative formula

The antiderivative of $\text{cos}(kx) $ is given by $\frac{1}{k}\text{sin}(kx) + C$.

Apply the constants

In the given problem, $k = 2$, so using the formula we get $\text{鈭珆 \text{cos}(2x) \, dx = \frac{1}{2} \text{sin}(2x) + C$.

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Key Concepts

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

Antiderivative

An antiderivative is a function whose derivative is the original function you started with. Think of it as the reverse process of differentiation. If you have a function, like $\text{cos}(kx)$, an antiderivative of this function is any function that, when differentiated, gives back $\text{cos}(kx)$.

For trigonometric functions, antiderivatives are particularly important because they allow us to solve integrals involving sine and cosine functions easily.

Integration

Integration is the process of finding the antiderivative of a function. It's essentially adding up small slices or areas to find the whole. In the context of indefinite integrals, you are trying to find a function that, when differentiated, gives the original function you started with.

In this exercise, we used an integration technique to find the antiderivative of $\text{cos}(2x)$. The integral of $\text{cos}(kx)$ is given by the formula:

$\text{鈭珆 \text{cos}(kx) \, dx = \frac{1}{k} \text{sin}(kx) + C$

where $C$ is the constant of integration and $k$ is a constant.

Trigonometric Functions

Trigonometric functions like sine and cosine are periodic functions that are very useful in many areas of math and science, including integrals.

For integration, it's helpful to remember some key antiderivatives:

Antiderivative of $ \text{cos}(x)$ is $ \text{sin}(x) + C $
Antiderivative of $ \text{sin}(x)$ is $-\text{cos}(x) + C$

When dealing with coefficients inside the trigonometric functions, like in $ \text{cos}(2x)$, we use the chain rule in reverse. So, if you integrate $ \text{cos}(2x)$, you get $ \frac{1}{2} \text{sin}(2x) + C $. This makes solving integrals involving trigonometric functions straightforward with the help of these rules.

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91影视

Find the following indefinite integrals. $$\int \cos 2 x \, d x$$

Short Answer

Step by step solution

Identify the integral form

Recall the antiderivative formula

Apply the constants

Key Concepts

Antiderivative

Integration

Trigonometric Functions

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