Evaluating Integrals Involving Radicals and Trigonometric Substitutions

Understanding how to evaluate integrals, especially those involving radicals, requires a solid grasp of various integration techniques. One such technique is the use of trigonometric substitutions. This article delves into the step-by-step process of evaluating the integral ∫ x/√(x^2 4) dx using a trigonometric substitution method. We will explore the reasoning behind each step, employ necessary calculus techniques, and conclude with a back-substitution to express the result in terms of the original variable.

Step 1: Substitution

When faced with integrals involving expressions of the form sqrt(x^2 a^2), a common approach is to use the substitution x a tan(θ). This substitution simplifies the expression under the radical.

For the given problem, set x 2 tan(θ). Consequently, the differential dx is rewritten as dx 2 sec^2(θ) dθ.

Step 2: Simplifying the Square Root

Substitute x 2 tan(θ) into the expression sqrt(x^2 4):

sqrt{x^2   4}  sqrt{(2 tan(θ))^2   4}  sqrt{4 tan^2(θ)   4}  sqrt{4(tan^2(θ)   1)}  2 sec(θ)

This simplification provides a simpler expression to work with in the integral.

Step 3: Substituting into the Integral

Now, substitute x and dx into the original integral:

∫ x/√(x^2   4) dx  ∫ (2 tan(θ)) / (2 sec(θ)) * 2 sec^2(θ) dθ  ∫ tan(θ) sec(θ) * 2 sec^2(θ) dθ  2 ∫ tan(θ) sec^3(θ) dθ

The integral has been simplified and is now in terms of θ.

Step 4: Integrating

Integrating tan(θ) sec^3(θ) requires a deeper understanding of trigonometric identities. Recognize that:

tan(θ) sec^3(θ)  (sin(θ) / cos(θ)) * (1 / cos^3(θ))  sin(θ) / cos^4(θ)

Using integration by parts or recognizing a pattern, the integral can be evaluated as:

∫ tan(θ) sec^3(θ) dθ  (1/2) sec^2(θ)   C

Thus, when we multiply by 2, we get:

2 ∫ tan(θ) sec^3(θ) dθ  sec^2(θ)   C

Step 5: Back-Substitution

To convert the result back into terms of x, recall the trigonometric identity:

sec(θ)  sqrt{1   tan^2(θ)}

Given that x 2 tan(θ), we have:

tan(θ)  x/2

Therefore:

sec^2(θ)  1   tan^2(θ)  1   (x/2)^2  1   x^2/4  (x^2   4)/4

Incorporating this back into our expression, we obtain:

sec^2(θ)   C  (x^2   4)/4   C

Thus, the final result of the integral is:

∫ x/√(x^2   4) dx  (x^2   4)/4   C

Summary

Evaluating integrals involving radicals, such as sqrt(x^2 4), can be systematically approached using trigonometric substitutions. By carefully performing each step and balancing the substitutions, we can successfully simplify the problem and obtain an expression in terms of the original variable.