Problem 15
Question
Find the general antiderivative. $$\int 5 \sec ^{2} x d x$$
Step-by-Step Solution
Verified Answer
The general antiderivative of \( 5 \sec ^{2} x \) is \( 5 \tan x + C \).
1Step 1: Identify Form and Apply Integral Rule
Identify that the \(\int \sec^2 x dx\) part of the given integral can be computed by using the known integral \(\int \sec^2 x dx = \tan x\). Apply this rule to the integral: \(\int 5 \sec^2 x dx\) = \(5 \int \sec^2 x dx\) = \(5 \tan x\).
2Step 2: Add Integration Constant
Add the constant of integration 'C' at the end to account for the 'lost' constant term during the differentiation process in the reverse order. So, our final result becomes \(5 \tan x + C\).
Key Concepts
Integration techniquesCalculus problem solvingTrigonometric integrals
Integration techniques
Integration techniques are fundamental methods used to solve integrals, which are the antiderivatives of functions. In calculus, integration is often seen as the reverse process of differentiation. To find the antiderivative of a given function, one typically employs a variety of techniques depending on the form of the function.
One of the most common techniques is **substitution**, which simplifies the integral by changing the variable to effectively make the problem easier to handle. Another is **integration by parts**, useful for integrals where the standard antiderivative rules do not suffice. Both techniques are powerful and are often taught extensively in calculus courses.
For special functions, like trigonometric integrals where standard antiderivatives are known, the process is often straightforward. For example, recognizing and using known formulas such as \( \int \sec^2 x \, dx = \tan x \) speeds up the integration process. In our current problem, we use this known integral to quickly solve for the antiderivative of \( 5 \sec^2 x \).
Remember, integrating a constant multiple is very straightforward: bring the constant outside the integral. Thus, for \( 5 \int \sec^2 x \ dx \), the constant 5 is simply multiplied after finding the integral of \( \sec^2 x \). This leads us directly to the solution, \( 5 \tan x + C \), where \( C \) is the constant of integration.
One of the most common techniques is **substitution**, which simplifies the integral by changing the variable to effectively make the problem easier to handle. Another is **integration by parts**, useful for integrals where the standard antiderivative rules do not suffice. Both techniques are powerful and are often taught extensively in calculus courses.
For special functions, like trigonometric integrals where standard antiderivatives are known, the process is often straightforward. For example, recognizing and using known formulas such as \( \int \sec^2 x \, dx = \tan x \) speeds up the integration process. In our current problem, we use this known integral to quickly solve for the antiderivative of \( 5 \sec^2 x \).
Remember, integrating a constant multiple is very straightforward: bring the constant outside the integral. Thus, for \( 5 \int \sec^2 x \ dx \), the constant 5 is simply multiplied after finding the integral of \( \sec^2 x \). This leads us directly to the solution, \( 5 \tan x + C \), where \( C \) is the constant of integration.
Calculus problem solving
Calculus problem solving is an essential skill that involves understanding concepts, applying known rules, and sometimes tackling unfamiliar problems creatively. In solving calculus problems, like finding antiderivatives, there are structured steps that help guide the process.
Start by analyzing the function you need to integrate. Look for patterns or known functions that have a standard antiderivative, such as \( \sec^2 x \) or \( \sin x \). This initial step of recognizing forms or patterns is crucial to applying the most effective integration technique.
Next, apply the appropriate rule or technique. If the integral involves a recognized form, like \( \sec^2 x \), use the direct rule: \( \int \sec^2 x \ dx = \tan x \). If not, decide whether substitution, integration by parts, or another method might simplify the problem.
Finally, remember to include the constant of integration \( C \). This constant is a reminder of the family of functions that could all have the same derivative. In exercises like these, always double-check your work by differentiating your answer to ensure it matches the original function under the integral.
Start by analyzing the function you need to integrate. Look for patterns or known functions that have a standard antiderivative, such as \( \sec^2 x \) or \( \sin x \). This initial step of recognizing forms or patterns is crucial to applying the most effective integration technique.
Next, apply the appropriate rule or technique. If the integral involves a recognized form, like \( \sec^2 x \), use the direct rule: \( \int \sec^2 x \ dx = \tan x \). If not, decide whether substitution, integration by parts, or another method might simplify the problem.
Finally, remember to include the constant of integration \( C \). This constant is a reminder of the family of functions that could all have the same derivative. In exercises like these, always double-check your work by differentiating your answer to ensure it matches the original function under the integral.
Trigonometric integrals
Trigonometric integrals involve integrating functions that are combinations of trigonometric terms. Understanding and identifying the integral forms of these functions make the process easier. Common trigonometric integrals include those of \( \sin x \), \( \cos x \), and \( \sec^2 x \).
One of the key approaches in dealing with trigonometric integrals is to leverage known integral forms. For instance, \( \int \sec^2 x \, dx = \tan x \) is a standard result used frequently in calculus problems.
When solving our given problem, \( \int 5 \sec^2 x \, dx \), recognizing the integral of \( \sec^2 x \) as \( \tan x \) allows us to simplify directly to \( 5 \tan x \). This technique saves time and avoids the need for lengthy substitution or rearrangement.
Moreover, it's important to understand the relationships and identities in trigonometry, such as the Pythagorean identities, which can be useful in more complex trigonometric integrals. Such knowledge enriches your toolbox, making you well-prepared for solving a variety of integral problems.
One of the key approaches in dealing with trigonometric integrals is to leverage known integral forms. For instance, \( \int \sec^2 x \, dx = \tan x \) is a standard result used frequently in calculus problems.
When solving our given problem, \( \int 5 \sec^2 x \, dx \), recognizing the integral of \( \sec^2 x \) as \( \tan x \) allows us to simplify directly to \( 5 \tan x \). This technique saves time and avoids the need for lengthy substitution or rearrangement.
Moreover, it's important to understand the relationships and identities in trigonometry, such as the Pythagorean identities, which can be useful in more complex trigonometric integrals. Such knowledge enriches your toolbox, making you well-prepared for solving a variety of integral problems.
Other exercises in this chapter
Problem 15
Use Riemann sums and a limit to compute the exact area under the curve. $$y=x^{2}+1 \text { on }[0,1]$$
View solution Problem 15
Use summation rules to compute the sum. $$\sum_{i=1}^{200}\left(4-3 i-i^{2}\right)$$
View solution Problem 16
Evaluate the indicated integral. $$\int \frac{\cos (1 / x)}{x^{2}} d x$$
View solution Problem 16
Evaluate the derivative using properties of logarithms where needed. $$\frac{d}{d x}(\ln \sqrt{\frac{x^{3}}{x^{5}+1}})$$
View solution