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Magazine Chapter 7: Problem 19
Evaluate the integral. $$ \int \frac{3 \sin x}{1+2 \cos x} d x $$
Short Answer
Expert verified
The integral evaluates to \( -\frac{3}{2} \ln|1 + 2 \cos x| + C \).
Step by step solution
01
Simplify the Integral Expression
Let's start by rewriting the given integral: \( \int \frac{3 \sin x}{1+2 \cos x} \, dx \). Notice that the integral involves \( \sin x \) and \( \cos x \). It may help to use a substitution to simplify the integration process.
02
Use an Appropriate Substitution
Set \( u = 1 + 2 \cos x \). Then, \( \frac{du}{dx} = -2 \sin x \). Solving for \( dx \), we get \( dx = -\frac{du}{2 \sin x} \). Substitute this into the integral, transforming it to \( \int \frac{3 \sin x}{u} \left(-\frac{du}{2 \sin x}\right) \).
03
Simplify and Resolve the Signs
The \( \sin x \) in the numerator and denominator cancel out. This leaves us with \( \int -\frac{3}{2} \frac{1}{u} \, du \). Factor out the constants to simplify further: \( -\frac{3}{2} \int \frac{1}{u} \, du \).
04
Integrate the Expression
The integral of \( \frac{1}{u} \) is \( \ln|u| \). So, \( -\frac{3}{2} \int \frac{1}{u} \, du = -\frac{3}{2} \ln|u| + C \), where \( C \) is the constant of integration.
05
Back-Substitute for \( u \)
We made the substitution \( u = 1 + 2 \cos x \) in Step 2. Substitute back to get the solution in terms of \( x \): \( -\frac{3}{2} \ln|1 + 2 \cos x| + C \).
06
Final Result
The evaluated integral is \( -\frac{3}{2} \ln|1 + 2 \cos x| + C \), which is our final answer.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Substitution
Trigonometric substitution is a powerful technique in integral calculus used to simplify integrals involving trigonometric functions like sine and cosine.
In our original problem, we simplify the given integral \[ \int \frac{3 \sin x}{1+2 \cos x} \, dx \]by making a substitution based on the structure of the denominator.
This involves identifying a substitution that will transform the integral into a simpler form.Let's break it down:
In our original problem, we simplify the given integral \[ \int \frac{3 \sin x}{1+2 \cos x} \, dx \]by making a substitution based on the structure of the denominator.
This involves identifying a substitution that will transform the integral into a simpler form.Let's break it down:
- In Step 2 of the solution, the substitution \( u = 1 + 2 \cos x \) is made because it simplifies the expression inside the integral. This substitution targets the complicated parts of the integrand influenced by cosine.
- When using trigonometric substitution, calculate the derivative, \( \frac{du}{dx} \), to replace \( dx \) in the integral. This step is crucial for correctly transforming the variable of integration.
- The derivatives help convert intricate trigonometric terms into simpler algebraic forms that are much easier to integrate.
Integration Techniques
Integrating functions, especially those involving trigonometric expressions, often requires a blend of techniques to solve.
In this problem, after applying \[ u = 1 + 2 \cos x \\text{and} \dx = -\frac{du}{2 \sin x}, \]we performed additional simplification by cancelling out \( \sin x \) in the processed integral.Simplification Steps to Learn:
In this problem, after applying \[ u = 1 + 2 \cos x \\text{and} \dx = -\frac{du}{2 \sin x}, \]we performed additional simplification by cancelling out \( \sin x \) in the processed integral.Simplification Steps to Learn:
- Factor out constants: They can simplify the integration process drastically. In this case, after canceling terms, the integral becomes \( -\frac{3}{2} \int \frac{1}{u} \, du \).
- Always check if the integral can be simplified further after substitution: Sometimes, recognizing simpler forms can help tackle the integration directly, as seen with the constant factor steps.
- Apply the basic forms of integrals: Here, the problem reduces to a base integral format of \( \log(u) \) which is straightforward to integrate.
Indefinite Integrals
Indefinite integrals, unlike definite integrals, yield a family of functions rather than a single numerical value.
The integral we solved:\[ -\frac{3}{2} \ln|1 + 2 \cos x| + C \]is an indefinite integral that includes a constant of integration, \( C \).Core Principles of Indefinite Integrals:
The integral we solved:\[ -\frac{3}{2} \ln|1 + 2 \cos x| + C \]is an indefinite integral that includes a constant of integration, \( C \).Core Principles of Indefinite Integrals:
- The constant \( C \) is essential: It represents the generality of the solution, indicating that there are infinitely many antiderivatives.
- Indefinite integrals involve finding anti-derivatives for functions: You're essentially reversing differentiation to find a function whose derivative gives back the original integrand.
- The process includes back-substituting to return to the original variable once integration is complete: This ensures the result is in terms of the variable initially presented in the integral.
