The arctangent function, often denoted as arctan or tan⁻¹, is a fundamental trigonometric function that represents the inverse of the tangent function. Finding the antiderivative, or integral, of the arctan function may seem daunting at first, but with the right approach, it can be tackled methodically.

In this comprehensive guide, we'll walk through the step-by-step process of finding the antiderivative of arctan, empowering students and enthusiasts alike to master this essential mathematical concept.

**A Brief Overview of Arctan Function**

Before delving into the antiderivative of arctan, it's essential to understand the properties of the arctan function. Arctan represents the angle whose tangent is a given number. In other words, if tan(x) = y, then arctan(y) = x. The arctan function has a domain of all real numbers and a range of (-π/2, π/2), making it a valuable tool in trigonometry and calculus.

**Using Integration by Parts**

Using integration by parts is a powerful method in calculus for evaluating integrals of products of functions. This technique involves breaking down the integral into two parts and applying a formula that relates the integral of the product of two functions to their derivatives. When dealing with the arctan function, we can express it as a product of simpler functions, such as x and 1/(1+x^2), and then apply integration by parts to find the antiderivative. This process allows us to decompose complex integrals and compute them more effectively, making integration by parts an invaluable tool in solving a wide range of mathematical problems.

**Step-by-Step Guide**

1. Identify the Integral: Start by identifying the integral of arctan(x) with respect to x.

2. Use Integration by Parts: Apply integration by parts, which states:

∫u dv = uv - ∫v du

Let u = arctan(x) and dv = dx. Then, differentiate u and integrate dv to obtain du and v, respectively.

3. Differentiate u: Differentiate u = arctan(x) with respect to x to find du.

d/dx(arctan(x)) = 1 / (1 + x^2)

4. Integrate dv: Integrate dv = dx to find v.

v = x

5. Apply Integration by Parts: Substitute the values of u, du, v, and dv into the integration by parts formula and evaluate the integral.

∫arctan(x) dx = x arctan(x) - ∫x / (1 + x^2) dx

6. Evaluate the Remaining Integral: The remaining integral can be evaluated using a trigonometric substitution or by employing partial fraction decomposition.

7. Simplify the Result: Once the integral is evaluated, simplify the result to obtain the final antiderivative of arctan.

**Practice and Application**

To solidify understanding and proficiency in finding the antiderivative of arctan, practice solving a variety of problems involving arctan functions. Utilize textbooks, online resources, and practice worksheets to access a diverse range of problems and scenarios. Additionally, apply the antiderivative of arctan in real-world contexts, such as physics, engineering, and finance, to understand its practical implications.

**Conclusion**

In conclusion, finding the antiderivative of arctan involves applying integration by parts and carefully evaluating the integral. By understanding the properties of the arctan function and mastering integration techniques, students can confidently tackle problems involving arctan expressions. Through practice, perseverance, and a systematic approach, mastering the antiderivative of arctan becomes an achievable goal, unlocking new opportunities for exploration and application in the realm of mathematics and beyond.