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Chapter 4: Problem 54

Find the indefinite integral. $$ \int \frac{\sec ^{2} 3 x}{4-\tan 3 x} d x $$

Short Answer

Expert verified
The indefinite integral of the given function is: \[ \int \frac{\sec^2 3x}{4-\tan 3x} dx = -\frac{1}{3} \ln |\tan 3x - 4| + C \]

Step by step solution

01

Identify the substitution

We can see that the given function involves a secant squared term and a tangent term. The derivative of the tangent function, in terms of x, is secant squared. So, we can choose the substitution \(u = \tan 3x\), with the differential \(du = 3 \sec^2 3x dx\).
02

Rewrite the function in terms of u

Now rewrite the integral using the substitution \(u = \tan 3x\): \[ \int \frac{\sec^2 3x}{4-\tan 3x} dx = \int \frac{1}{4-u} \cdot \frac{1}{3} du \]
03

Integrate the function

Now we can integrate the function with respect to u: \[ \frac{1}{3} \int \frac{1}{4-u}du = -\frac{1}{3} \int \frac{1}{u-4} du \] We recognize the integral as the natural logarithm (ln) and integrate it: \[ -\frac{1}{3} \int \frac{1}{u-4}du = -\frac{1}{3} \ln |u-4| + C \]
04

Substitute back in terms of x

Now we substitute back the original expression for u in terms of x: \[ -\frac{1}{3} \ln |u-4| + C = -\frac{1}{3} \ln |\tan 3x - 4| + C \] So the indefinite integral of the given function is: \[ \int \frac{\sec^2 3x}{4-\tan 3x} dx = -\frac{1}{3} \ln |\tan 3x - 4| + C \]

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Key Concepts

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

Substitution Method
The substitution method is an integration technique used to simplify complex integrals by transforming variables. It's especially useful when dealing with composite functions. To perform substitution, identify part of the integral that, if replaced, would simplify the integration process. Steps to Perform Substitution:
  • Identify a suitable substitution, often a function within the integral whose derivative is also present.
  • Declare the substitution, such as \( u = g(x) \), and find the corresponding differential \( du = g'(x) \, dx \).
  • Rewrite the original integral in terms of \( u \). This step often involves expressing \( dx \) in terms of \( du \).
  • Integrate with respect to \( u \).
  • Once the integration is complete, revert back to the original variable \( x \) by replacing \( u \) with its expression in \( x \).
In our exercise, we replaced \( u = \tan 3x \), leveraging the derivative \( du = 3 \sec^2 3x \, dx \). This substitution turned a complex integral into a simpler one that could then be easily integrated.
Indefinite Integral
An indefinite integral, often referred to simply as an antiderivative, is a fundamental concept in calculus that represents a family of functions whose derivative is the given function. The result of finding an indefinite integral is a function plus a constant of integration, \( C \). Key Characteristics:
  • Unlike definite integrals, indefinite integrals do not have limits of integration.
  • The result includes a constant \( C \) because differentiation of any constant is zero.
  • Indefinite integrals are expressed with the integral symbol \( \int \) followed by the function and differential, such as \( \int f(x) \, dx \).
In the original exercise, finding the indefinite integral \( \int \frac{\sec^2 3x}{4-\tan 3x} \, dx \) involved recognizing it could be transformed into an integration with respect to \( u \). The output included \( C \) to account for the constant term present in all antiderivatives.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It is a crucial function not only in integration but also in various fields of mathematics and the sciences. Understanding the Natural Logarithm in Integrals:
  • One of the most common integrals involving the natural logarithm is \( \int \frac{1}{x} \, dx = \ln |x| + C \).
  • The presence of the absolute value ensures the domain of \( \ln \) does not include negative values, making the antiderivative well-defined.
  • In integration, the natural logarithm often appears when dealing with expressions of the form \( \frac{1}{u} \), triggering substitution.
In our exercise, the antiderivative \( \int \frac{1}{4-u} \, du \) turned into \( -\frac{1}{3} \ln |u-4| + C \) after substitution, showcasing the natural logarithm's role in integral calculus.

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