Integral ∫ 0 Π / 2 Arctan ⁡ ( 2 Tan ⁡ 2 ( X ) ) D X \int_{0}^{\pi/2} \arctan \left(2\tan^{2}\left(x\right)\right) \mathrm{d}x ∫ 0 Π /2 ​ Arctan ( 2 Tan 2 ( X ) ) D X

Introduction

In this article, we will delve into the evaluation of a definite integral involving the arctan function. The given integral is $\int_{0}^{\pi/2} \arctan \left(2\tan^{2}\left(x\right)\right) \mathrm{d}x$. This integral may seem straightforward, but it requires a careful approach to evaluate it correctly. We will use various techniques from real analysis and calculus to prove the given result.

Background and Motivation

The arctan function is a fundamental function in mathematics, and it has numerous applications in various fields, including calculus, algebra, and geometry. The definite integral of the arctan function is a crucial concept in real analysis, and it has been extensively studied by mathematicians. In this article, we will focus on evaluating the definite integral of the arctan function with a specific argument.

The Given Integral

The given integral is $\int_{0}^{\pi/2} \arctan \left(2\tan^{2}\left(x\right)\right) \mathrm{d}x$. This integral involves the arctan function with a specific argument, which is $2\tan^{2}\left(x\right)$. The integral is taken over the interval $\left[0, \frac{\pi}{2}\right]$. Our goal is to evaluate this integral and prove the given result.

Using Trigonometric Substitution

To evaluate the given integral, we can use the trigonometric substitution method. We will substitute $x = \arctan\left(t\right)$, which implies that $\tan\left(x\right) = t$. This substitution will simplify the integral and make it easier to evaluate.

Let $x = \arctan\left(t\right)$. Then, we have $\tan\left(x\right) = t$. We can rewrite the integral in terms of $t$ as follows:

$$\int_{0}^{\pi/2} \arctan \left(2\tan^{2}\left(x\right)\right) \mathrm{d}x = \int_{0}^{1} \arctan \left(2t^{2}\right) \frac{1}{1+t^{2}} \mathrm{d}t$$

Evaluating the Integral

Now, we can evaluate the integral using the substitution method. We will integrate the arctan function with respect to $t$ and then substitute back to the original variable $x$.

Let $u = 2t^{2}$. Then, we have $\mathrm{d}u = 4t \mathrm{d}t$. We can rewrite the integral in terms of $u$ as follows:

$$\int_{0}^{1} \arctan \left(2t^{2}\right) \frac{1}{1+t^{2}} \mathrm{d}t = \frac{1}{4} \int_{0}^{2} \arctan\left(u\right) \frac{1}{1+\frac{u^{2}}{4}} \mathrm{d}u$$

Simplifying the Integral

Now, we can simplify the integral by combining the fractions in the denominator.

$$\frac{1}{4} \int_{0}^{2} \arctan\left(u\right) \frac{1}{1+\frac{u^{2}}{4}} \mathrm{d}u = \frac{1}{4} \int_{0}^{2} \arctan\left(u\right) \frac{4}{4+u^{2}} \mathrm{d}u$$

Evaluating the Simplified Integral

Now, we can evaluate the simplified integral using the arctan function properties.

$$\frac{1}{4} \int_{0}^{2} \arctan\left(u\right) \frac{4}{4+u^{2}} \mathrm{d}u = \frac{1}{4} \left[ \arctan\left(u\right) \ln\left(4+u^{2}\right) \right]_{0}^{2}$$

Substituting Back

Now, we can substitute back to the original variable $x$.

$$\frac{1}{4} \left[ \arctan\left(u\right) \ln\left(4+u^{2}\right) \right]_{0}^{2} = \frac{1}{4} \left[ \arctan\left(2\right) \ln\left(4+4\right) - \arctan\left(0\right) \ln\left(4+0^{2}\right) \right]$$

Simplifying the Result

Now, we can simplify the result by evaluating the arctan and ln functions.

$$\frac{1}{4} \left[ \arctan\left(2\right) \ln\left(4+4\right) - \arctan\left(0\right) \ln\left(4+0^{2}\right) \right] = \frac{1}{4} \left[ \arctan\left(2\right) \ln\left(8\right) - 0 \ln\left(4\right) \right]$$

Final Result

Now, we can simplify the result by evaluating the arctan and ln functions.

$$\frac{1}{4} \left[ \arctan\left(2\right) \ln\left(8\right) - 0 \ln\left(4\right) \right] = \frac{1}{4} \left[ \arctan\left(2\right) \ln\left(2^{3}\right) \right]$$

Using the Property of Logarithm

Now, we can use the property of logarithm to simplify the result.

$$\frac{1}{4} \left[ \arctan\left(2\right) \ln\left(2^{3}\right) \right] = \frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right]$$

Simplifying the Result

Now, we can simplify the result by evaluating the arctan function.

$$\frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right] = \frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right]$$

Using the Property of Arctan

Now, we can use the property of arctan to simplify the result.

$$\frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right] = \frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right]$$

Final Result

Now, we can simplify the result by evaluating the arctan function.

$$\frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right] = \frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right]$$

Conclusion

In this article, we evaluated the definite integral of the arctan function with a specific argument. We used various techniques from real analysis and calculus to prove the given result. The final result is $\frac{1}{4} \left[ \arctan\left(2\right) 3\ln\left(2\right) \right]$. This result can be further simplified using the properties of arctan and logarithm functions.

Final Answer

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Introduction

In our previous article, we evaluated the definite integral of the arctan function with a specific argument. We used various techniques from real analysis and calculus to prove the given result. In this article, we will answer some frequently asked questions related to the evaluation of the definite integral of the arctan function.

Q: What is the arctan function?

A: The arctan function, also known as the inverse tangent function, is a mathematical function that returns the angle whose tangent is a given number. It is denoted by $\arctan(x)$ and is the inverse of the tangent function.

Q: What is the definite integral of the arctan function?

A: The definite integral of the arctan function is a mathematical expression that represents the area under the curve of the arctan function over a given interval. It is denoted by $\int_{a}^{b} \arctan(x) \mathrm{d}x$ and is a fundamental concept in real analysis and calculus.

Q: How do you evaluate the definite integral of the arctan function?

A: To evaluate the definite integral of the arctan function, you can use various techniques from real analysis and calculus, such as substitution, integration by parts, and trigonometric substitution. In our previous article, we used the trigonometric substitution method to evaluate the definite integral of the arctan function with a specific argument.

Q: What is the final result of the definite integral of the arctan function?

A: The final result of the definite integral of the arctan function is $\frac{\pi}{4} \arctan\left(\frac{1}{2}\right)$. This result can be further simplified using the properties of arctan and logarithm functions.

Q: What are some common applications of the definite integral of the arctan function?

A: The definite integral of the arctan function has numerous applications in various fields, including calculus, algebra, and geometry. Some common applications include:

• Evaluating the area under curves
• Finding the volume of solids
• Solving differential equations
• Modeling real-world phenomena

Q: How do you use the definite integral of the arctan function in real-world applications?

A: To use the definite integral of the arctan function in real-world applications, you can apply the techniques and methods learned in calculus and real analysis. For example, you can use the definite integral of the arctan function to:

• Model the growth of populations
• Analyze the behavior of electrical circuits
• Optimize the design of mechanical systems
• Solve problems in physics and engineering

Q: What are some common mistakes to avoid when evaluating the definite integral of the arctan function?

A: Some common mistakes to avoid when evaluating the definite integral of the arctan function include:

• Failing to use the correct substitution or integration method
• Making errors in the calculation of the integral
• Failing to simplify the result using the properties of arctan and logarithm functions
• Not checking the validity of the result

Conclusion

In this article, we answered some frequently asked questions related to the evaluation of the definite integral of the arctan function. We hope that this article has provided valuable insights and information for students and professionals in the field of mathematics and science.

Final Answer

The final answer is $\boxed{\frac{\pi}{4} \arctan\left(\frac{1}{2}\right)}$.