Integral Of 1 Sqrt X 2 A 2

The Indefinite Integral of 1/√(x² + a²)

The integral ∫1/√(x² + a²) dx is a classic example of an integral that requires a specific technique to solve. It doesn't yield to simple u-substitution, and integration by parts isn't immediately obvious. Instead, we'll employ a trigonometric substitution, a powerful method for tackling integrals involving quadratic expressions. This article will delve into the solution, exploring the steps involved, the underlying principles, and some of its applications.

Understanding Trigonometric Substitution

Trigonometric substitution relies on the fundamental trigonometric identities to simplify expressions containing quadratic terms. The key is to choose a substitution that transforms the quadratic expression into a trigonometric identity, allowing for simpler integration. The choice of substitution depends on the form of the quadratic:

• For √(a² - x²): Use x = a sin θ
• For √(a² + x²): Use x = a tan θ
• For √(x² - a²): Use x = a sec θ

In our case, we have √(x² + a²), so we'll use the substitution x = a tan θ.

Step-by-Step Solution

1. Substitution: Let x = a tan θ. Then, dx = a sec² θ dθ.

2. Substitute into the Integral: Substituting x and dx into our integral, we get:

∫ 1/√(x² + a²) dx = ∫ 1/√((a tan θ)² + a²) * a sec² θ dθ

3. Simplify the Expression: Recall the trigonometric identity: 1 + tan² θ = sec² θ. Using this, we can simplify the expression under the square root:

√((a tan θ)² + a²) = √(a²(tan² θ + 1)) = √(a² sec² θ) = a sec θ (assuming a > 0)

Therefore, our integral becomes:

∫ 1/√((a tan θ)² + a²) * a sec² θ dθ = ∫ (a sec² θ) / (a sec θ) dθ = ∫ sec θ dθ

4. Integrate sec θ: The integral of sec θ is a standard integral that requires a clever trick. We multiply and divide by (sec θ + tan θ):

∫ sec θ dθ = ∫ sec θ * (sec θ + tan θ) / (sec θ + tan θ) dθ

Let u = sec θ + tan θ. Then, du = (sec θ tan θ + sec² θ) dθ = sec θ (tan θ + sec θ) dθ = sec θ * u dθ.

This allows us to rewrite the integral as:

∫ (1/u) du = ln|u| + C

5. Back-Substitution: Substitute back u = sec θ + tan θ:

ln|sec θ + tan θ| + C

6. Express in terms of x: We need to express the result in terms of the original variable, x. Recall that x = a tan θ. We can construct a right-angled triangle with opposite side x, adjacent side a, and hypotenuse √(x² + a²). From this triangle, we can deduce:

• tan θ = x/a
• sec θ = √(x² + a²)/a

Substituting these back into our result:

ln|√(x² + a²)/a + x/a| + C

7. Final Simplification: We can simplify this further:

ln| (√(x² + a²) + x) / a | + C = ln|√(x² + a²) + x| - ln|a| + C

Since 'a' is a constant and C is an arbitrary constant of integration, we can combine -ln|a| with C, resulting in a new constant C₁:

ln|√(x² + a²) + x| + C₁

This is the final solution for the indefinite integral of 1/√(x² + a²).

Verifying the Solution

We can verify the solution using differentiation. Differentiating ln|√(x² + a²) + x| + C₁ with respect to x:

d/dx [ln|√(x² + a²) + x| + C₁] = 1/(√(x² + a²) + x) * (1/(2√(x² + a²)) * 2x + 1)

= 1/(√(x² + a²) + x) * (x/√(x² + a²) + 1)

= (x + √(x² + a²)) / (√(x² + a²) + x) * (1/√(x² + a²))

= 1/√(x² + a²)

This confirms our solution is correct.

Applications

The integral ∫1/√(x² + a²) dx has numerous applications in various fields, including:

• Physics: Calculating the gravitational potential of a distributed mass.
• Engineering: Solving problems involving cable sag and catenary curves.
• Geometry: Finding arc lengths of curves.
• Calculus: Demonstrating techniques of integration and substitution.

Further Exploration: Definite Integrals

While we've focused on the indefinite integral, understanding the definite integral is also crucial. Evaluating the definite integral involves substituting the limits of integration into the antiderivative. For example, to evaluate ∫ from 0 to 1 1/√(x² + a²) dx, you would substitute x=1 and x=0 into ln|√(x² + a²) + x| + C₁ and find the difference.

Dealing with Different Forms

It's important to note that the form of the quadratic significantly impacts the chosen substitution. Understanding the differences between √(a² - x²), √(a² + x²), and √(x² - a²) is crucial for applying the correct trigonometric substitution. Practice is key to mastering this technique.

Conclusion

The integral ∫1/√(x² + a²) dx showcases the power and elegance of trigonometric substitution. By carefully choosing a substitution and applying trigonometric identities, we can transform a seemingly intractable integral into a manageable one. Understanding this process not only enhances your calculus skills but also opens the door to solving a variety of problems in different scientific and engineering disciplines. The detailed step-by-step breakdown, coupled with the verification and application examples, should equip you with a thorough understanding of this essential integration technique. Remember to practice regularly to solidify your understanding and develop proficiency in solving similar integrals.