Evaluating The Integral Of 1 Over 5 + X Squared
In this article, we will delve into the evaluation of the integral $\int \frac{1}{5+x^2} dx$. This integral falls under the category of standard integrals that can be solved using trigonometric substitution. Specifically, it resembles the form $\int \frac{1}{a^2 + x^2} dx$, which has a well-known solution. We will explore the step-by-step process of solving this integral, highlighting the underlying principles and techniques involved. Understanding these techniques is crucial for tackling more complex integration problems in calculus and related fields. This article aims to provide a comprehensive guide, making the solution accessible to students and enthusiasts alike. By the end of this discussion, you will be equipped with the knowledge to solve similar integrals and appreciate the elegance of integration methods.
We are tasked with evaluating the definite integral: $\int \frac{1}{5+x^2} dx$ This integral is a classic example that can be solved using a trigonometric substitution. The form of the integral suggests that we can use the substitution $x = a \tan(\theta)$, where $a$ is a constant. By doing so, we can transform the integral into a form that is easier to solve. The key to this method is recognizing the similarity between the denominator $5 + x^2$ and the trigonometric identity $1 + \tan^2(\theta) = \sec^2(\theta)$. This recognition allows us to simplify the integral and express it in terms of trigonometric functions. The subsequent steps involve performing the substitution, simplifying the integral, and then integrating with respect to $\theta$. Finally, we will convert back to the original variable $x$ to obtain the final solution. This problem is not only a fundamental exercise in calculus but also a demonstration of the power and versatility of trigonometric substitutions in solving integrals.
The method of trigonometric substitution is a powerful technique for evaluating integrals involving expressions of the form $a^2 + x^2$, $a^2 - x^2$, or $x^2 - a^2$. In our case, we have the expression $5 + x^2$, which closely resembles $a^2 + x^2$. The goal is to make a substitution that simplifies the integral using trigonometric identities. For the expression $a^2 + x^2$, the appropriate substitution is $x = a \tan(\theta)$. This substitution allows us to utilize the identity $1 + \tan^2(\theta) = \sec^2(\theta)$. By substituting $x = a \tan(\theta)$ into the integral, we can transform the denominator into a form that can be simplified using this identity. The success of this method hinges on choosing the correct trigonometric substitution and correctly applying the relevant identities. This technique is widely used in calculus and is an essential tool for solving a variety of integrals. Understanding the rationale behind trigonometric substitution and its application is crucial for mastering integration techniques.
To solve the integral $\int \frac{1}{5+x^2} dx$, we will use the method of trigonometric substitution. Here's a step-by-step breakdown:
Step 1: Identify the appropriate substitution
We observe that the integral has the form $\int \frac1}{a^2 + x^2} dx$, where $a^2 = 5$. Thus, $a = \sqrt{5}$. The appropriate trigonometric substitution is \tan(\theta)$. This substitution is chosen because it allows us to utilize the trigonometric identity $1 + \tan^2(\theta) = \sec^2(\theta)$, which will simplify the integral.
Step 2: Calculate dx
Differentiating $x$ with respect to $\theta$, we get: $\frac{dx}{d\theta} = \sqrt{5} \sec^2(\theta)$ Therefore, $dx = \sqrt{5} \sec^2(\theta) d\theta$. This expression for $dx$ is crucial for substituting into the integral, as it replaces the differential $dx$ with an expression in terms of $\theta$ and $d\theta$.
Step 3: Substitute into the integral
Substituting $x = \sqrt{5} \tan(\theta)$ and $dx = \sqrt{5} \sec^2(\theta) d\theta$ into the integral, we have:
Step 4: Simplify the integral
Simplify the denominator: $5 + (\sqrt{5} \tan(\theta))^2 = 5 + 5 \tan^2(\theta) = 5(1 + \tan^2(\theta))$. Using the trigonometric identity $1 + \tan^2(\theta) = \sec^2(\theta)$, we get: $5(1 + \tan^2(\theta)) = 5 \sec^2(\theta)$. Now substitute this back into the integral:
Simplify further: $\int \frac{\sqrt{5}}{5} d\theta = \frac{1}{\sqrt{5}} \int d\theta$
Step 5: Integrate with respect to θ
Integrating with respect to $\theta$, we obtain: $\frac{1}{\sqrt{5}} \int d\theta = \frac{1}{\sqrt{5}} \theta + C$, where $C$ is the constant of integration. This step is straightforward, as the integral of $d\theta$ is simply $\theta$.
Step 6: Convert back to the original variable x
We need to express $\theta$ in terms of $x$. From the substitution $x = \sqrt{5} \tan(\theta)$, we have $\tan(\theta) = \frac{x}{\sqrt{5}}$. Therefore, $\theta = \arctan(\frac{x}{\sqrt{5}})$. Substituting this back into our result, we get:
Thus, the final solution is: $\int \frac{1}{5+x^2} dx = \frac{1}{\sqrt{5}} \arctan(\frac{x}{\sqrt{5}}) + C$
An alternative method to solve this integral is by using the standard integral formula for the arctangent function. The integral we are evaluating, $\int \frac1}{5+x^2} dx$, is a special case of the general form $\int \frac{1}{a^2 + x^2} dx$, where $a$ is a constant. The standard integral formula for this form is{a^2 + x^2} dx = \frac{1}{a} \arctan(\frac{x}{a}) + C$, where $C$ is the constant of integration. This formula is derived from the derivative of the arctangent function and is a fundamental result in calculus. Using this formula can often provide a more direct route to the solution, especially when the integral matches the standard form closely. In our case, recognizing the direct application of this formula simplifies the process, avoiding the need for a full trigonometric substitution. This approach highlights the importance of recognizing standard integral forms and utilizing them to efficiently solve integration problems.
Applying the Formula
In our integral, $\int \frac1}{5+x^2} dx$, we can identify $a^2 = 5$, so $a = \sqrt{5}$. Applying the standard integral formula, we directly get{5+x^2} dx = \frac{1}{\sqrt{5}} \arctan(\frac{x}{\sqrt{5}}) + C$ This method is significantly quicker as it bypasses the trigonometric substitution steps. The key is to recognize the integral's form and apply the appropriate standard formula. This approach reinforces the idea that recognizing patterns and applying known formulas can greatly simplify the process of integration. It's a valuable skill for anyone studying calculus, as it allows for efficient problem-solving and a deeper understanding of the underlying principles of integration.
In summary, we have successfully evaluated the integral $\int \frac1}{5+x^2} dx$ using two different methods{\sqrt{5}} \arctan(\frac{x}{\sqrt{5}}) + C$. The trigonometric substitution method involves making a substitution that simplifies the integral using trigonometric identities, while the standard integral formula provides a direct solution by recognizing the form of the integral. The choice of method often depends on the problem and personal preference. However, understanding both methods provides a more comprehensive approach to solving integrals of this type. This exercise demonstrates the versatility of integration techniques and the importance of recognizing standard forms to simplify the process. Mastering these techniques is crucial for further studies in calculus and related fields. The ability to solve such integrals is a fundamental skill for anyone working with mathematical models and engineering applications.
