Problem 62
Question
Right, or wrong? Say which for each formula and give a brief reason for each answer. $$ \begin{array}{l}{\text { a. } \int \tan \theta \sec ^{2} \theta d \theta=\frac{\sec ^{3} \theta}{3}+C} \\ {\text { b. } \int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \tan ^{2} \theta+C} \\ {\text { c. } \int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \sec ^{2} \theta+C}\end{array} $$
Step-by-Step Solution
Verified Answer
Only the third formula is correct; it's the result of the integral.
1Step 1: Identify the Given Integral
We need to evaluate the integral \( \int \tan \theta \sec^2 \theta \, d\theta \). This is the same integral in all the expressions provided.
2Step 2: Perform the Integration
To solve \( \int \tan \theta \sec^2 \theta \, d\theta \), we recognize that the derivative of \( \sec \theta \) is \( \sec \theta \tan \theta \). So, we apply the substitution \( u = \sec \theta \), which gives \( du = \sec \theta \tan \theta \, d\theta \). Therefore, \( \int \tan \theta \sec^2 \theta \, d\theta = \frac{1}{2} u^2 + C = \frac{1}{2} \sec^2 \theta + C \).
3Step 3: Evaluate First Formula
The first formula asserts \( \int \tan \theta \sec^2 \theta \, d\theta = \frac{\sec^3 \theta}{3} + C \). This is incorrect because our integration provided \( \frac{1}{2} \sec^2 \theta + C \).
4Step 4: Evaluate Second Formula
The second formula asserts \( \int \tan \theta \sec^2 \theta \, d\theta = \frac{1}{2} \tan^2 \theta + C \). This is incorrect because it should be \( \frac{1}{2} \sec^2 \theta + C \).
5Step 5: Evaluate Third Formula
The third formula asserts \( \int \tan \theta \sec^2 \theta \, d\theta = \frac{1}{2} \sec^2 \theta + C \). This is correct as it matches our integration result.
Key Concepts
Trigonometric IntegralsSubstitution MethodDefinite and Indefinite Integrals
Trigonometric Integrals
Trigonometric integrals involve functions like sine, cosine, tangent, and their reciprocals. These integrals frequently appear in calculus problems because trigonometric functions are periodic and have important applications in fields such as physics, engineering, and mathematics. For integrals involving products or powers of trigonometric functions, recognizing identities and derivatives is key.
In the exercise, we had to integrate \( \int \tan \theta \sec^2 \theta \, d\theta \). This falls under the category of trigonometric integrals, specifically because it involves a product of tangent and secant squared. By identifying derivatives of trigonometric functions, like \( \frac{d}{d\theta} [\sec \theta] = \sec \theta \tan \theta \), we can simplify the integration process. Using the right identities makes the integration much simpler, helping us reach the solution efficiently, which is an essential skill for solving calculus problems.
In the exercise, we had to integrate \( \int \tan \theta \sec^2 \theta \, d\theta \). This falls under the category of trigonometric integrals, specifically because it involves a product of tangent and secant squared. By identifying derivatives of trigonometric functions, like \( \frac{d}{d\theta} [\sec \theta] = \sec \theta \tan \theta \), we can simplify the integration process. Using the right identities makes the integration much simpler, helping us reach the solution efficiently, which is an essential skill for solving calculus problems.
Substitution Method
The substitution method is a common and powerful technique used in calculus to simplify integrals by making substitutions that reduce complexity. Essentially, this involves changing the variable of integration to transform the original integral into a simpler form.
In our exercise, we used the substitution \( u = \sec \theta \) to solve \( \int \tan \theta \sec^2 \theta \, d\theta \). With this substitution, we also find \( du = \sec \theta \tan \theta \, d\theta \). The substitution then simplifies the integral to \( \int u \, du \), which is straightforward to integrate, resulting in \( \frac{1}{2} u^2 + C \). Finally, by substituting back \( u = \sec \theta \), the integral becomes \( \frac{1}{2} \sec^2 \theta + C \).
Using substitution effectively requires recognizing the function and its derivative within the integral. This allows for reducing the problem to a simpler form, making integration manageable and error-free.
In our exercise, we used the substitution \( u = \sec \theta \) to solve \( \int \tan \theta \sec^2 \theta \, d\theta \). With this substitution, we also find \( du = \sec \theta \tan \theta \, d\theta \). The substitution then simplifies the integral to \( \int u \, du \), which is straightforward to integrate, resulting in \( \frac{1}{2} u^2 + C \). Finally, by substituting back \( u = \sec \theta \), the integral becomes \( \frac{1}{2} \sec^2 \theta + C \).
Using substitution effectively requires recognizing the function and its derivative within the integral. This allows for reducing the problem to a simpler form, making integration manageable and error-free.
Definite and Indefinite Integrals
Integrals in calculus come in two main types: definite and indefinite. Indefinite integrals, like those in our exercise, are essentially antiderivatives and include an arbitrary constant \( C \) because there are infinitely many functions whose derivatives can equal the integrand. These are represented as \( \int f(x) \, dx = F(x) + C \).
In contrast, definite integrals have specific limits of integration, giving a numerical result that represents the area under the curve defined by \( f(x) \) between two bounds. They are given by \( \int_a^b f(x) \, dx \) and do not include the constant \( C \).
Understanding both types of integrals is crucial in calculus. Indefinite integrals capture the general form of antiderivatives, while definite integrals provide specific numerical areas. Both are foundational to various applications in mathematical modeling and real-world problem-solving.
In contrast, definite integrals have specific limits of integration, giving a numerical result that represents the area under the curve defined by \( f(x) \) between two bounds. They are given by \( \int_a^b f(x) \, dx \) and do not include the constant \( C \).
Understanding both types of integrals is crucial in calculus. Indefinite integrals capture the general form of antiderivatives, while definite integrals provide specific numerical areas. Both are foundational to various applications in mathematical modeling and real-world problem-solving.
Other exercises in this chapter
Problem 61
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Right, or wrong? Say which for each formula and give a brief reason for each answer. $$ \begin{array}{l}{\text { a. } \int(2 x+1)^{2} d x=\frac{(2 x+1)^{3}}{3}+
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