Question

In the following exercises, find each indefinite integral by using appropriate substitutions. $$ \int \frac{\ln (\sin x)}{\tan x} d x $$

Short answer

The integral is \( \frac{(\ln(\sin x))^2}{2} + C \).

Step-by-step solution

Identify Appropriate Substitution

Notice that the integrand involves a natural logarithm and a trigonometric function. A suitable substitution would simplify the expression. Let us choose the substitution \( u = \ln(\sin x) \) because its derivative features within the integral in the form compatible with \( \tan x \).

Compute the Derivative of u

Now compute the derivative \( du \). If \( u = \ln(\sin x) \), then \( du = \frac{1}{\sin x} \cdot \cos x \cdot dx = \cot x \, dx \). Thus, \( dx = \frac{du}{\cot x} = \frac{du}{\frac{1}{\tan x}} = du \cdot \tan x \).

Rewrite the Integral in Terms of u

Substituting \( u \) and \( dx = du \cdot \tan x \) into the original integral gives us: \[ \int \frac{u}{\tan x} \tan x \, du = \int u \, du. \]

Integrate with respect to u

Now, integrate \( \int u \, du \). This is a straightforward antiderivative: \[ \int u \, du = \frac{u^2}{2} + C. \]

Substitute Back for x

Recall that \( u = \ln(\sin x) \). Substituting back, we get:\[ \frac{(\ln(\sin x))^2}{2} + C. \]

Key concepts

Substitution Method

The substitution method is a fundamental technique for solving indefinite integrals. It involves transforming the original variable in the integrand to a new variable that simplifies the integration process. In simple terms, you change the variables in order to make the integral easier to solve.

In the given exercise, the substitution method is utilized by choosing \( u = \ln(\sin x) \). This substitution is selected because its derivative, \( \cot x \, dx \), is conveniently present in the integral. By converting the variables, you simplify the integration to a simpler form. After substitution, you integrate with respect to \( u \), which often leads to a more recognizable and solvable integral. Finally, once you find the antiderivative, you substitute back to the original variable to get the answer.

Trigonometric Integrals

Integrals that involve trigonometric functions are referred to as trigonometric integrals. These can often be quite complex due to the periodic and oscillating nature of trigonometric functions. Knowing certain identities and relationships among these functions can make solving these integrals more manageable.

For instance, in the exercise, we see \( \tan x \) and \( \sin x \) within the integral. Understanding the properties of these functions, such as \( \cot x = \frac{1}{\tan x} \), you can manipulate the expression to reveal potential substitutions or simplifications. This understanding is crucial to setting up the correct substitution and streamlining the integration process.

Integration Techniques

When tackling indefinite integrals, using a variety of integration techniques enhances your problem-solving toolkit. Techniques like substitution, integration by parts, and partial fraction decomposition allow you to solve a wide range of integrals efficiently.

The exercise example predominantly uses the substitution method; however, recognizing when other techniques might be necessary is important. Integration often starts with identifying parts of the integrand that resemble derivatives of familiar functions or that can be rewritten in a more compatible form for the known techniques. This strategic approach ensures you can handle a variety of complex integrals with confidence.