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Chapter 28: Problem 24

Integrate each of the given functions. $$\int \cos x \ln (\sin x) d x$$

Short Answer

Expert verified
The integral is \( \ln(\sin x) \sin x - x + C \).

Step by step solution

01

Identify the Integration Technique

To solve the integral \( \int \cos x \ln (\sin x) \, dx \), we notice that integration by parts is suitable here. This is because we have a product of functions: \( \cos x \) and \( \ln(\sin x) \).
02

Choose Functions for Integration by Parts

Integration by parts formula is \( \int u \, dv = uv - \int v \, du \). Let \( u = \ln(\sin x) \) and \( dv = \cos x \, dx \). Then, we find \( du \) and \( v \).
03

Compute Derivatives and Integrals

Differentiate \( u \) to get \( du = \frac{1}{\sin x} \cos x \, dx = \cot x \, dx \). Integrate \( dv \) to get \( v = \sin x \) (since \( \int \cos x \, dx = \sin x \)).
04

Apply Integration by Parts Formula

Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:\[ \int \cos x \ln(\sin x) \, dx = \ln(\sin x) \cdot \sin x - \int \sin x \cdot \cot x \, dx. \]
05

Simplify the Resulting Integral

The integral \( \int \sin x \cdot \cot x \, dx \) simplifies to \( \int \frac{\sin x}{\tan x} \, dx = \int dx \) because \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \) and thus \( \frac{\sin x}{\tan x} = \cos x \). Hence, the integral simplifies to \( x \).
06

Complete the Integration

Substitute back to get:\[ \int \cos x \ln(\sin x) \, dx = \ln(\sin x) \cdot \sin x - x + C, \]where \( C \) is the constant of integration.

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Key Concepts

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

Integration by Parts
Integration by parts is a technique used for integrating products of functions. It transforms the product of two functions into simpler, more manageable terms.
The formula is:
  • \( \int u \, dv = uv - \int v \, du \)
In our example, we have \( \cos x \ln(\sin x) \), which fits this scheme perfectly because it's a product of functions. The key to using this technique is to identify parts of the integral as \( u \) and \( dv \).
For example:
  • Choose \( u = \ln(\sin x) \), making \( dv = \cos x \, dx \).
  • Calculate \( du \) by differentiating \( u \), resulting in \( du = \cot x \, dx \).
  • Integrate \( dv \) to obtain \( v = \sin x \).
Finally, substitute into the formula to simplify the integral step by step.
Trigonometric Integrals
Trigonometric integrals often involve functions like sine, cosine, tangent, etc. These functions can intertwine, making integration challenging.
To tackle such integrals:
  • Know the basic integral and derivative forms, as they frequently appear in different variations.
  • Use identities such as \( \sin^2 x + \cos^2 x = 1 \) to simplify expressions.
  • In our exercise, the integral involves \( \cos x \ln(\sin x) \), which simplifies through further breakdown \( \int \sin x \cdot \cot x \, dx \).
This transforms to \( \int dx \), making it easier to integrate. Understanding these identities and transformations is crucial in solving trigonometric integrals.
Mathematical Notation
Mathematical notation serves as the language for expressing complex mathematical concepts and operations succinctly.
Here are some notations seen in the solution:
  • \( \int \) represents the integral sign, indicating an integration process.
  • \( dx \) is the differential, indicating the variable of integration.
  • \( \ln(x) \) represents the natural logarithm, highlighting the specific function to be integrated.
  • \( C \) denotes the constant of integration, crucial for representing indefinite integrals.
Using precise notation provides clarity and precision, allowing complex solutions to be presented accurately and compactly. It's a common language shared by mathematicians to communicate ideas and solutions effectively.

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Most popular questions from this chapter

Solve the given problems by integration. In the study of the lifting force \(L\) due to a stream of fluid passing around a cylinder, the equation \(L=k \int_{0}^{2 \pi}\left(a \sin \theta+b \sin ^{2} \theta-b \sin ^{3} \theta\right) d \theta\) is used. Here, \(k, a,\) and \(b\) are constants and \(\theta\) is the angle from the direction of flow. Evaluate the integral.

Solve the given problems by integration. Perform the integration \(\int x \sqrt{1-x^{2}} d x\) (a) by using the power formula, and (b) by trigonometric substitution. Compare results.

Identify the form of each integral as being inverse sine. inverse tangent, logarithmic, or general power, as in Examples 6 and 7. Do not integrate. In each part (a), explain how the choice was made. (a) \(\int \frac{2 x d x}{\sqrt{4-9 x^{2}}}\) (b) \(\int \frac{2 d x}{\sqrt{4-9 x^{2}}}\) (c) \(\int \frac{2 d x}{4-9 x}\)

Integrate each of the given functions. $$\int \frac{\sec ^{2} t \tan t}{4+\sec ^{2} t} d t$$

Solve the given problems by integration. Explain how to integrate \(\int \frac{d x}{\sqrt{x}(1+x)} .\) What is the result?
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