Problem 49
Question
Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int \tan ^{2} 3 x d x$$
Step-by-Step Solution
Verified Answer
Question: Evaluate the indefinite integral of the function, $$\int \tan^2 3x dx.$$
Answer: The indefinite integral is given by $$\int \tan^2 3x dx = \frac{1}{3} (\tan 3x - 3x) + C.$$
1Step 1: Recall the trigonometric identity for tangent squared
Recall that tangent squared can be expressed in terms of secant squared using the trigonometric identity
$$\tan^2 x = \sec^2 x - 1$$
For the given integral, we have:
$$\int \tan^2 3x dx = \int (\sec^2 3x - 1) dx$$
2Step 2: Apply substitution
Let's perform the substitution \(u = 3x\), then \(du = 3 dx\) and \(dx = \frac{1}{3} du\).
Now we have:
$$\int (\sec^2 3x - 1) dx = \frac{1}{3} \int (\sec^2 u - 1) du$$
3Step 3: Evaluate the integral
Now, we can evaluate the integral:
$$\frac{1}{3} \int (\sec^2 u - 1) du = \frac{1}{3} \left(\int \sec^2 u du - \int 1 du \right)$$
The integral of \(\sec^2 u\) is \(\tan u\) and the integral of \(1\) is \(u\). Thus, we get:
$$\frac{1}{3} \left(\int \sec^2 u du - \int 1 du \right) = \frac{1}{3} (\tan u - u) + C$$
4Step 4: Back-substitute
Finally, we substitute back \(u=3x\):
$$\frac{1}{3} (\tan 3x - 3x) + C$$
So the final answer is:
$$\int \tan^2 3x dx = \frac{1}{3} (\tan 3x - 3x) + C$$
Key Concepts
Trigonometric IdentitiesSubstitution Method in IntegrationComputer Algebra SystemAntiderivatives
Trigonometric Identities
Trigonometric identities are fundamental tools in calculus, particularly when dealing with integrals and derivatives involving trigonometric functions. These identities allow us to simplify complex expressions and solve trigonometric equations.
For instance, a highly useful identity is \[ \tan^2x = \sec^2x - 1 \.\] This identity expresses the tangent function in terms of the secant function, which can be advantageous as it often leads to more straightforward integrals. When integrating functions involving \( \tan^2x \), this identity can transform the integral into more manageable terms, such as \[ \int \tan^2x \, dx = \int (\sec^2x - 1) \, dx \.\] Recognizing when to apply these identities is key to streamlining the integration process.
For instance, a highly useful identity is \[ \tan^2x = \sec^2x - 1 \.\] This identity expresses the tangent function in terms of the secant function, which can be advantageous as it often leads to more straightforward integrals. When integrating functions involving \( \tan^2x \), this identity can transform the integral into more manageable terms, such as \[ \int \tan^2x \, dx = \int (\sec^2x - 1) \, dx \.\] Recognizing when to apply these identities is key to streamlining the integration process.
Substitution Method in Integration
The substitution method is a technique used to evaluate more complex integrals by simplifying them into basic forms. It involves making a substitution to convert the integral into one that is easier to manage.
Take the substitution \( u = 3x \), which leads to \( du = 3 dx \) and consequently \( dx = \frac{1}{3} du \). When applied to our integral, it changes the original equation to a form with \( u \) as the variable. Here's the substitution applied to our problem: \[ \int (\sec^2 3x - 1) dx = \frac{1}{3} \int (\sec^2 u - 1) du \.\] Employing the substitution method streamlines the integration process and can be especially handy when working with trigonometric functions.
Take the substitution \( u = 3x \), which leads to \( du = 3 dx \) and consequently \( dx = \frac{1}{3} du \). When applied to our integral, it changes the original equation to a form with \( u \) as the variable. Here's the substitution applied to our problem: \[ \int (\sec^2 3x - 1) dx = \frac{1}{3} \int (\sec^2 u - 1) du \.\] Employing the substitution method streamlines the integration process and can be especially handy when working with trigonometric functions.
Computer Algebra System
A Computer Algebra System (CAS) is a software tool that facilitates the manipulation and solution of mathematical expressions symbolically. These systems are designed to perform operations such as solving equations, simplifying expressions, and calculating derivatives and integrals.
Utilizing a CAS to evaluate integrals can significantly simplify the process, providing exact solutions and serving as a valuable learning aid. In our exercise, a CAS could be used to confirm the steps and the final answer. When used correctly, CAS tools help students to understand underlying mathematical principles and to check their manual calculations for accuracy.
Utilizing a CAS to evaluate integrals can significantly simplify the process, providing exact solutions and serving as a valuable learning aid. In our exercise, a CAS could be used to confirm the steps and the final answer. When used correctly, CAS tools help students to understand underlying mathematical principles and to check their manual calculations for accuracy.
Antiderivatives
Antiderivatives, also known as indefinite integrals, represent the reverse process of differentiation. They are used to find a function when its derivative is given. The antiderivative of a function is not unique because it includes an arbitrary constant \( C \), representing the family of all antiderivatives of the function.
For example, the antiderivative of \( \sec^2u \) is \( \tan u \), and the antiderivative of 1 with respect to \( u \) is \( u \). Thus, we can find the antiderivative of \( \sec^2u - 1 \) as: \[ \int (\sec^2 u - 1) du = \tan u - u + C \.\] Identifying antiderivatives is a central aspect of integration, and it enables the calculation of areas, volumes, and other physical quantities.
For example, the antiderivative of \( \sec^2u \) is \( \tan u \), and the antiderivative of 1 with respect to \( u \) is \( u \). Thus, we can find the antiderivative of \( \sec^2u - 1 \) as: \[ \int (\sec^2 u - 1) du = \tan u - u + C \.\] Identifying antiderivatives is a central aspect of integration, and it enables the calculation of areas, volumes, and other physical quantities.
Other exercises in this chapter
Problem 49
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