Solving Integral Of Root 1 + X Squared A Comprehensive Guide

In this article, we will delve into the solution of the integral ∫√(1 + x²) dx, a classic problem in calculus that often appears in mathematics exams and textbooks. This integral is a prime example of a problem that requires the use of trigonometric substitution, a powerful technique for simplifying integrals involving square roots of quadratic expressions. Understanding this integral not only enhances your calculus skills but also provides a foundation for tackling more complex problems in physics, engineering, and other scientific disciplines. We will explore the step-by-step process of solving this integral, providing a detailed explanation of each step to ensure clarity and comprehension. Whether you are a student preparing for an exam, a teacher looking for a clear explanation, or simply a math enthusiast, this article will provide you with a comprehensive understanding of this integral and its solution.

The problem we aim to solve is the indefinite integral of the square root of (1 + x²) with respect to x. Mathematically, this is represented as:

∫√(1 + x²) dx

This integral does not have a straightforward solution using basic integration rules. The presence of the square root and the sum of a constant and a variable term necessitates a more advanced technique, specifically trigonometric substitution. By employing trigonometric substitution, we can transform the integral into a more manageable form, allowing us to apply standard integration formulas and simplify the expression. The goal is to find a function whose derivative is √(1 + x²), and this process involves careful manipulation and application of trigonometric identities.

Trigonometric substitution is a method used to simplify integrals involving expressions of the form √(a² - x²), √(a² + x²), or √(x² - a²). In our case, we have √(1 + x²), which fits the form √(a² + x²) where a = 1. The appropriate trigonometric substitution for this form is x = a tan θ, which in our case simplifies to x = tan θ. This substitution is crucial because it allows us to eliminate the square root and express the integral in terms of trigonometric functions, which are often easier to integrate. The key idea is to use the Pythagorean trigonometric identity 1 + tan² θ = sec² θ to simplify the expression under the square root.

Step-by-Step Solution

1. Substitute x = tan θ

Let x = tan θ. This substitution implies that dx = sec² θ dθ. We choose this substitution because it aligns with the form √(a² + x²) where a = 1. The derivative of tan θ is sec² θ, which is a standard result in calculus and is essential for the substitution process. This step transforms the integral from an algebraic form to a trigonometric form, which is a crucial step in simplifying the integral.

2. Rewrite the Integral

Substitute x = tan θ and dx = sec² θ dθ into the integral:

∫√(1 + x²) dx = ∫√(1 + tan² θ) * sec² θ dθ

This substitution replaces x with tan θ and dx with sec² θ dθ. The next step involves using the trigonometric identity 1 + tan² θ = sec² θ to simplify the expression under the square root. This step is critical as it transforms the integral into a form that can be easily handled using trigonometric identities and standard integration techniques.

3. Apply the Trigonometric Identity

Using the identity 1 + tan² θ = sec² θ, we get:

∫√(sec² θ) * sec² θ dθ = ∫sec θ * sec² θ dθ = ∫sec³ θ dθ

This step simplifies the square root, resulting in sec θ. The integral now becomes the integral of sec³ θ, which is a standard integral but requires a specific technique to solve. The simplification to ∫sec³ θ dθ is a significant milestone in the solution process.

4. Integrate ∫sec³ θ dθ

The integral of sec³ θ is a classic integral that is typically solved using integration by parts. We can write ∫sec³ θ dθ as ∫sec θ * sec² θ dθ and apply integration by parts.

Let u = sec θ and dv = sec² θ dθ. Then, du = sec θ tan θ dθ and v = tan θ.

Using integration by parts formula, ∫u dv = uv - ∫v du:

∫sec³ θ dθ = sec θ tan θ - ∫tan θ * sec θ tan θ dθ

∫sec³ θ dθ = sec θ tan θ - ∫sec θ tan² θ dθ

This step involves applying the integration by parts formula, which is a fundamental technique in calculus. The choice of u and dv is crucial for simplifying the integral. The resulting integral ∫sec θ tan² θ dθ still needs further simplification.

5. Simplify the Integral

Use the identity tan² θ = sec² θ - 1:

∫sec θ tan² θ dθ = ∫sec θ (sec² θ - 1) dθ = ∫sec³ θ dθ - ∫sec θ dθ

Substituting this back into the integration by parts result:

∫sec³ θ dθ = sec θ tan θ - (∫sec³ θ dθ - ∫sec θ dθ)

This step uses the trigonometric identity tan² θ = sec² θ - 1 to break down the integral into simpler terms. The resulting expression includes ∫sec³ θ dθ, which is the original integral, and ∫sec θ dθ, which is a standard integral. This step is crucial for isolating the integral we want to solve.

6. Solve for ∫sec³ θ dθ

Move the ∫sec³ θ dθ term to the left side:

2∫sec³ θ dθ = sec θ tan θ + ∫sec θ dθ

The integral of sec θ is a standard result: ∫sec θ dθ = ln|sec θ + tan θ| + C

So, 2∫sec³ θ dθ = sec θ tan θ + ln|sec θ + tan θ| + C

Divide by 2:

∫sec³ θ dθ = (1/2)sec θ tan θ + (1/2)ln|sec θ + tan θ| + C

This step involves algebraic manipulation to isolate ∫sec³ θ dθ and substituting the known integral of sec θ. The constant of integration, C, is added to account for all possible antiderivatives.

7. Substitute Back for x

Since x = tan θ, we can draw a right triangle to find sec θ. If tan θ = x = x/1, then the opposite side is x, the adjacent side is 1, and the hypotenuse is √(1 + x²).

Thus, sec θ = √(1 + x²).

Substitute back into the integral:

∫√(1 + x²) dx = (1/2)√(1 + x²) * x + (1/2)ln|√(1 + x²) + x| + C

This step is the reverse of the initial substitution. We use the relationship x = tan θ to express the solution in terms of x. The right triangle is a helpful tool for determining the trigonometric functions in terms of x.

8. Final Solution

Therefore, the integral ∫√(1 + x²) dx is:

(x/2)√(1 + x²) + (1/2)ln(x + √(1 + x²)) + C

This is the final solution to the integral. It represents the antiderivative of √(1 + x²) with respect to x. The solution includes both an algebraic term and a logarithmic term, reflecting the complexity of the original integral.

In conclusion, we have successfully solved the integral ∫√(1 + x²) dx using the method of trigonometric substitution. This problem highlights the importance of choosing the correct substitution and applying trigonometric identities to simplify complex integrals. The step-by-step solution provided in this article offers a clear and comprehensive understanding of the process, making it a valuable resource for students and educators alike. Mastering such techniques is crucial for tackling more advanced problems in calculus and related fields.

The final solution, (x/2)√(1 + x²) + (1/2)ln(x + √(1 + x²)) + C, is a testament to the power of trigonometric substitution in simplifying seemingly intractable integrals. By understanding and practicing this method, you can enhance your calculus skills and gain confidence in solving a wide range of integration problems. Remember to always check your solution by differentiating it to ensure it matches the original integrand.