Problem 29
Question
Evaluate the indefinite integral. \(\int \sec ^{3} x \tan x d x\)
Step-by-Step Solution
Verified Answer
The integral is \( \frac{\sec^3 x}{3} + C \).
1Step 1: Identify the Function
Identify the function that you are integrating: \[ f(x) = \sec^3 x \tan x \] This is the function inside the integral.
2Step 2: Recognize Derivative within Integral
Notice that the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), which is part of the function being integrated. This indicates that substitution might be a good strategy.
3Step 3: Choose Substitution
Let \( u = \sec(x) \). Then, the derivative \( du = \sec(x) \tan(x) \, dx \) gives us exactly the differential component in the integral: \( du = \sec(x) \tan(x) \, dx \).
4Step 4: Substitution
Substitute \( u = \sec(x) \) and \( du = \sec(x) \tan(x) \, dx \) into the integral:\[ \int u^2 \, du \]
5Step 5: Integrate
Integrate \( u^2 \) with respect to \( u \):\[ \int u^2 \, du = \frac{u^3}{3} + C \] where \( C \) is the constant of integration.
6Step 6: Back-substitute
Replace \( u \) with \( \sec(x) \) to return to the variable \( x \):\[ \frac{\sec^3(x)}{3} + C \]
7Step 7: Conclusion
The indefinite integral of \( \sec^3 x \tan x \) is given by:\[ \int \sec^3 x \tan x \, dx = \frac{\sec^3 x}{3} + C \]
Key Concepts
Integration by SubstitutionTrigonometric IntegralsIntegral Calculus
Integration by Substitution
Integration by substitution is a powerful method used in calculus to simplify integrals and make them easier to solve. It's akin to reverse chain rule in differentiation. The goal is to identify a part of the integral that can be replaced with a new variable, often simplifying the integrand in the process.
In this exercise, we observed that the integrand is \( \sec^3 x \tan x \). We realize that the differentiation of \( \sec(x) \) gives \( \sec(x) \tan(x) \), which suggests substitution as a suitable approach. When we set \( u = \sec(x) \), it makes the integral much easier to handle.
The differential \( du \) equals \( \sec(x) \tan(x) \, dx \), perfectly matching a part of our integrand. This makes our integral change from a complex trigonometric integral to a simple polynomial in terms of \( u \). This technique not only simplifies integration but also allows for a more intuitive understanding of how different parts of the function are related.
In this exercise, we observed that the integrand is \( \sec^3 x \tan x \). We realize that the differentiation of \( \sec(x) \) gives \( \sec(x) \tan(x) \), which suggests substitution as a suitable approach. When we set \( u = \sec(x) \), it makes the integral much easier to handle.
The differential \( du \) equals \( \sec(x) \tan(x) \, dx \), perfectly matching a part of our integrand. This makes our integral change from a complex trigonometric integral to a simple polynomial in terms of \( u \). This technique not only simplifies integration but also allows for a more intuitive understanding of how different parts of the function are related.
Trigonometric Integrals
Trigonometric integrals frequently appear in calculus when functions involving trigonometric terms need integration. These integrals can sometimes be dealt with through substitution, identities, or even transformations.
In the given exercise, the integrand \( \sec^3 x \tan x \) is a trigonometric form where substitution made things clear. Recognizing derivatives of trigonometric functions can help identify the correct substitution. For instance, knowing that \( \frac{d}{dx} [ \sec(x) ] = \sec(x) \tan(x) \) was crucial.
Trigonometric identities, like \( \sec^2(x) = 1 + \tan^2(x) \), sometimes simplify these problems, but in this particular case, substitution was more straightforward to apply. Mastery of these techniques allows you to handle a wide variety of trigonometric integrals you might encounter in calculus.
In the given exercise, the integrand \( \sec^3 x \tan x \) is a trigonometric form where substitution made things clear. Recognizing derivatives of trigonometric functions can help identify the correct substitution. For instance, knowing that \( \frac{d}{dx} [ \sec(x) ] = \sec(x) \tan(x) \) was crucial.
Trigonometric identities, like \( \sec^2(x) = 1 + \tan^2(x) \), sometimes simplify these problems, but in this particular case, substitution was more straightforward to apply. Mastery of these techniques allows you to handle a wide variety of trigonometric integrals you might encounter in calculus.
Integral Calculus
Integral calculus is a branch of calculus concerned with the concept of integration. It deals with finding the antiderivative or the area under curves, and often requires transforming a complex integrand into a more manageable form.
The purpose of indefinite integration, as seen in this exercise, is to find a function whose derivative is the original integrand. This provides a broader solution than definite integration, which gives a specific numerical value.
The purpose of indefinite integration, as seen in this exercise, is to find a function whose derivative is the original integrand. This provides a broader solution than definite integration, which gives a specific numerical value.
- Antiderivatives: The goal in integral calculus is often to compute the antiderivative. In our problem, we achieved this by transforming \( \sec^3 x \tan x \) to \( u^2 \), simplifying our task.
- Substitution Method: As demonstrated, substitution is a vital technique in integral calculus, aiding in handling complex integrals by changing variables.
- Constant of Integration: The presence of \( C \) in our final solution \( \frac{\sec^3 x}{3} + C \) represents the general solution, highlighting the integral's indefinite nature.
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