Problem 25 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{8 d x}{\left(4 x^{2}+1\right)^{2}} $$

Step-by-Step Solution

Answer

The integral evaluates to \( 4 \tan^{-1}(2x) + \frac{4x}{1 + 4x^2} + C \).

Step 1: Identify the Method

We need to evaluate the integral \( \int \frac{8 \, dx}{(4x^2 + 1)^2} \). This integral can be solved using the technique of trigonometric substitution, recognizing the denominator as a form that resembles \( a^2 + x^2 \).

Step 2: Use Trigonometric Substitution

Since the expression \( 4x^2 + 1 \) resembles \( a^2 + x^2 \), let \( x = \frac{1}{2} \tan(\theta) \). This means \( dx = \frac{1}{2} \sec^2(\theta) \, d\theta \). The expression \( 4x^2 + 1 \) becomes \( \sec^2(\theta) \) under this substitution.

Step 3: Substitute and Simplify

Substitute \( x \) and \( dx \) into the integral: \[ \int \frac{8 \, \frac{1}{2} \sec^2(\theta) \, d\theta}{(\sec^2(\theta))^2} = \int 8 \cos^2(\theta) \, d\theta. \]The \( \sec^2(\theta) \) terms cancel out, simplifying to \( 8 \cos^2(\theta) \).

Step 4: Solve the Integral using a Trigonometric Identity

Use the trigonometric identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) to solve the integral:\[ \int 8 \cos^2(\theta) \, d\theta = 4 \int (1 + \cos(2\theta)) \, d\theta. \]Integrate:\[ 4 \left( \theta + \frac{1}{2} \sin(2\theta) \right) + C. \]

Step 5: Substitute Back to x

Recall \( x = \frac{1}{2} \tan(\theta) \), then \( \theta = \tan^{-1}(2x) \). Thus, substitute back:\[ 4 \left( \tan^{-1}(2x) + \frac{1}{2} \sin(2\theta) \right) + C. \]Since \( \sin(2\theta) = \frac{2x}{1 + 4x^2} \), substitute this back:\[ 4 \tan^{-1}(2x) + \frac{4x}{1 + 4x^2} + C. \]

Step 6: Final Expression

Combine the terms to get the final expression:\[ 4 \tan^{-1}(2x) + \frac{4x}{1 + 4x^2} + C. \]This is the evaluated integral of the given function.

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of calculus concerned with the accumulation of quantities and the areas under and between curves. It focuses on finding the integral of functions, which is essentially the reverse process of differentiation. This allows us to calculate not only areas and volumes but also a variety of other physical quantities. In essence, when you find an integral, you are looking for a function whose derivative matches the given integrand.
In this exercise, we are tasked with evaluating the integral \( \int \frac{8 \, dx}{(4x^2+1)^2} \). This integral requires the use of advanced techniques, such as trigonometric substitution, because the direct application of basic integral rules is not sufficient.
Understanding these concepts is crucial because they extend the ability to solve complex integrals that cannot be easily handled by simpler techniques like basic substitution or integration by parts.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true for all values of the variables involved. They play a critical role in calculus and precalculus, especially when simplifying expressions and solving integrals. Common trigonometric identities include Pythagorean identities, angle sum identities, and double angle identities among others.
During the solution process of this integral, the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \) is used, which is a power-reduction formula. This identity helps simplify the integral by reducing the power of cosine, making it easier to solve.
Knowing and recognizing these identities can significantly simplify integration problems, particularly when dealing with trigonometric functions, and enhance your understanding of the structure of trigonometric expressions in calculus.

Integration Techniques

Integration techniques refer to the various methods used to evaluate integrals, which are often complex expressions. Some integral calculus problems require straightforward methods like basic substitution, but others demand more sophisticated strategies. These techniques include integration by parts, partial fraction decomposition, and trigonometric substitution, among others.
In this exercise, trigonometric substitution is used because the integral \( \int \frac{8 \, dx}{(4x^2+1)^2} \) has a denominator resembling the form \( a^2 + x^2 \). By letting \( x = \frac{1}{2} \tan(\theta) \), the substitution transforms the expression into one involving trigonometric functions, which can be further simplified and integrated. This technique is particularly useful for integrals involving square roots and quadratic expressions.
Trigonometric substitution is a powerful technique that builds on the connection between algebraic and trigonometric functions. Mastering these integration techniques expands your toolkit for tackling a wide array of integrals encountered in calculus.