Question

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$ \int \tan (2 x) d x $$

Short answer

\( \int \tan(2x) \, dx = -\frac{1}{2} \ln|\cos(2x)| + C \)

Step-by-step solution

Choose Appropriate Substitution

To solve the integral \( \int \tan(2x) \, dx \), we start by using the identity for tangent in terms of sine and cosine: \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \). This suggests a substitution involving \( u = \cos(2x) \), leading to \( du = -2\sin(2x) \, dx \). Thus, \( dx = -\frac{du}{2\sin(2x)} \).

Modify Integral with Substitution

Substitute so that the integral becomes \( \int \frac{\sin(2x)}{u} \left(-\frac{du}{2\sin(2x)}\right) = -\frac{1}{2} \int \frac{1}{u} \, du \). The \( \sin(2x) \) terms cancel out.

Integrate with Respect to u

Recognize that \( \int \frac{1}{u} \, du \) integrates to \( \ln|u| \). Therefore, the integral is \( -\frac{1}{2} \ln|u| + C \), where \( C \) is the integration constant.

Substitute Back to Original Variable

Since \( u = \cos(2x) \), substitute back to the original variable: \( -\frac{1}{2} \ln|\cos(2x)| + C \).

Final Result

Thus, the final result of the original integral is \( \int \tan(2x) \, dx = -\frac{1}{2} \ln|\cos(2x)| + C \).

Key concepts

Substitution Method

The substitution method is a popular technique used in calculus to simplify the process of integration. It is particularly helpful when dealing with trigonometric integrals. The main idea is to replace a complex expression with a single variable, which transforms the integral into a simpler form.Consider the integral \( \int \tan(2x) \, dx \). We can rewrite \( \tan(2x) \) using the identity \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \). By observing this, we note that a substitution involving \( u = \cos(2x) \) could be effective. This is because the derivative, \( du = -2\sin(2x) \, dx \), naturally aligns with the structure of the integral.

• Step-by-step substitution: Replace \( \cos(2x) \) with \( u \) and solve for \( dx \) in terms of \( du \): \( dx = -\frac{du}{2\sin(2x)} \).
• Simplify: The integral then simplifies to \( -\frac{1}{2} \int \frac{1}{u} \, du \).

This method streamlines the integration by reducing the original problem to a basic integral in terms of \( u \), which can be more easily evaluated.

Integration Techniques

Integration techniques are methods used to find solutions to integrals that are not immediately obvious. When dealing with trigonometric integrals, like \( \int \tan(2x) \, dx \), these techniques are crucial to simplifying and solving the problem.In our example, we started with a trigonometric function \( \tan(2x) \), which we converted using the identity \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \). To solve this, we employed a substitution, which is a common technique to tackle integrals involving trigonometric functions.

• Using identities: Trigonometric identities transform complicated functions into simpler forms.
• Substitution: We deduced the need for substitution by recognizing the derivative \( du = -2\sin(2x) \, dx \) already in the integral.
• Recognizing integrals: Identify basic integrals within the expression, like \( \int \frac{1}{u} \, du \), which simplifies the process.

These techniques help manage complex integrals by breaking them down into more familiar forms that we can easily handle.

Logarithmic Integration

Logarithmic integration is a method used when you encounter integrals that involve natural logarithms. This approach is particularly useful when dealing with integrals of the form \( \int \frac{1}{x} \, dx \), which is known to integrate to \( \ln|x| \).In our given problem, after using substitution, the integral reduces to \( -\frac{1}{2} \int \frac{1}{u} \, du \). This is a classic example of logarithmic integration, where we end up integrating \( \frac{1}{u} \) to get \( \ln|u| \).

• Identify and integrate: Recognizing the form \( \int \frac{1}{u} \, du \) allows us to apply the natural logarithm immediately.
• Substitute back: Once integrated, we substitute the original expressions back in place of \( u \). Here, \( u = \cos(2x) \), giving the final expression \( -\frac{1}{2} \ln|\cos(2x)| + C \).

This technique simplifies the task by leveraging the properties of logarithms to solve what might otherwise be a complex integral.