Question

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

Short answer

The integral evaluates to \(-\frac{(\ln(\cos x))^2}{2} + C\).

Step-by-step solution

Identify the Substitution

First, look at the integral \( \int \ln(\cos x) \tan x \, dx \). To simplify this integral, notice that if we set \( u = \ln(\cos x) \), we can find a substitution related to \( \tan x \).

Differentiate u

Differentiate \( u = \ln(\cos x) \). Using the chain rule, we have \( \frac{du}{dx} = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x \). Therefore, \( du = -\tan x \, dx \).

Substitute and Integrate

We can substitute \( du = -\tan x \, dx \) into the original integral: \( \int \ln(\cos x) \tan x \, dx = -\int u \, du \).

Evaluate the New Integral

The integral \( -\int u \, du \) is a simple power rule integration problem: \( -\int u \, du = -\frac{u^2}{2} + C \), where \( C \) is the constant of integration.

Back-Substitute to Original Variable

Replace \( u \) with \( \ln(\cos x) \) to get back to the original variables: \( -\frac{(\ln(\cos x))^2}{2} + C \). Thus, the evaluated integral is \( -\frac{(\ln(\cos x))^2}{2} + C \).

Key concepts

Logarithmic Integration

Logarithmic integration deals with integrals that involve logarithmic functions. In our exercise, the function inside the integral is the natural logarithm, denoted by \( \ln(\cos x) \). When integrating logarithmic functions, we often look for ways to assign a suitable substitution to simplify the integral into a basic form we are familiar with. This often involves recognizing how derivatives or transformations involving logarithms can help. In our case, the substitution builds on the relationship between logarithmic functions and their derivatives to streamline the integration process. This type of integral can sometimes be transformed into a straightforward expression using the power rule integration, once the logarithmic component is aptly handled.

Substitution Method

The substitution method is an essential technique in calculus used to simplify integrals. By substituting a portion of the integrand (the function being integrated) with a single variable, the integral transforms into a simpler problem. In the exercise, we substitute \( u = \ln(\cos x) \). This simplifies the integral into terms of \( u \).
For substitution, we also need to differentiate \( u \) to express \( dx \) in terms of \( du \). When we differentiate \( u \), we find \( du = -\tan x \, dx \), thereby allowing us to replace \( \tan x \, dx \) in the original integral with \( -du \). This substitution reduces the original problem into an easily solvable one.

Chain Rule

The chain rule is a vital concept in calculus, useful when differentiating compositions of functions. For instance, if a function \( u(x) \) is composed of another function, the derivative is found by multiplying the derivative of the outer function by the derivative of the inner function.
In solving our integral, the chain rule helps differentiate \( u = \ln(\cos x) \). By applying the chain rule, we find \( \frac{du}{dx} = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x \). This connection reveals an important relationship that allows our substitution method to proceed smoothly. By effectively using the chain rule, the trigonometric nature of the function transforms into an algebraic form that simplifies integration.

Integration Techniques

There are several integration techniques applied to solve complex integrals, and choosing the right one depends on the function's structure. One common technique is substitution, as we've seen applied to this integral. This converts a complex problem into a simpler task. Another technique is integrating by parts, which can sometimes provide an alternative path when substitution isn't enough.
Once substitution reduces the integral to \( -\int u \, du \), we can straightforwardly apply the power rule. The power rule states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = 1 \), simplifying the integral to \( -\frac{u^2}{2} + C \). Converting back to our original variable, we return to \( -\frac{(\ln(\cos x))^2}{2} + C \), concluding the integration process. Such techniques allow us to address challenges posed by more complex integrals across different functions.