Problem 402 - Calculus Volume 1

Question

In the following exercises, find each indefinite integral, using appropriate substitutions. $$\int \frac{d x}{|x| \sqrt{4 x^{2}-16}}$$

Step-by-Step Solution

Answer

The indefinite integral is \( \sqrt{1 - \frac{4}{x^2}} + C \).

Step 1: Simplify under the square root

Start by simplifying the expression under the square root, \( \sqrt{4x^2 - 16} \). Notice that you can factor out a 4: \( \sqrt{4(x^2 - 4)} = \sqrt{4} \cdot \sqrt{x^2 - 4} = 2\sqrt{x^2 - 4} \). So the integral becomes \( \frac{1}{|x|} \cdot \frac{1}{2\sqrt{x^2 - 4}} \).

Step 2: Choose an appropriate substitution

Recognize that \( x^2 - 4 \) is in the form suitable for a trigonometric substitution, specifically \( x = 2 \sec(\theta) \). This gives \( dx = 2\sec(\theta)\tan(\theta) \, d\theta \). Substitute these into the integral.

Step 3: Substitute and simplify the integral

The integral becomes \( \int \frac{1}{|2\sec(\theta)|} \cdot \frac{2\sec(\theta)\tan(\theta)}{2\sqrt{(2\sec(\theta))^2 - 4}} \, d\theta \). Simplifying using the Pythagorean identity \( (2\sec(\theta))^2 - 4 = 4\sec^2(\theta) - 4 \) reduces to \( 4 \tan^2(\theta) \), and \( 2\tan(\theta) \) is the square root, the integral becomes \( \int \frac{2\sec(\theta)\tan(\theta)}{2|2\tan(\theta)|} \, d\theta \).

Step 4: Further simplify the expression

Simplify the fraction by noticing that the \( 2\sec(\theta)\tan(\theta) \) in the numerator cancels out the terms in the denominator, resulting in \( \int \frac{1}{|\sec(\theta)|} \, d\theta \). This simplifies to \( \int |\cos(\theta)| \, d\theta \).

Step 5: Solve the integral

Assume \( \theta \) is in the first quadrant. Then the magnitude function is irrelevant, and we solve \( \int \cos(\theta) \, d\theta = \sin(\theta) + C \), where \( C \) is the constant of integration.

Step 6: Back substitute the variable x

Revert \( \theta \) back to \( x \) using \( x = 2\sec(\theta) \), where \( \sec(\theta) = \frac{x}{2} \). Therefore, \( \cos(\theta) = \frac{2}{x} \) and \( \sin(\theta) = \sqrt{1 - \frac{4}{x^2}} \). Substitute back to get the original variable \( x \).

Step 7: Final expression substitution

The final expression becomes \( \int \frac{1}{|x|\sqrt{4x^2 - 16}} \, dx = \sin(\theta) + C = \sqrt{1 - \frac{4}{x^2}} + C \) after substituting back using trigonometric identities.

Key Concepts

Trigonometric Substitution

When dealing with integrals involving square roots, especially those of the form \( a^2 - x^2 \), trigonometric substitution is a nifty technique. This method relies on the relationships in right triangles and trigonometric identities to simplify the integral. For example, an expression like \( x^2 - 4 \) can be managed by using \( x = 2 \sec(\theta) \) for calculation simplification. This transformation utilizes the identity \( \sec^2(\theta) = 1 + \tan^2(\theta) \). After substituting, the integral is re-expressed in terms of \( \theta \), disposing of the square root hurdles and often turning the integral into a more straightforward trigonometric integral. This approach not only simplifies computation but also often allows for a more direct path to the solution.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry, allowing us to substitute complex expressions with simpler forms. In the context of our integral problem, the expression \( 4\sec^2(\theta) - 4 \) simplifies using this identity to \( 4\tan^2(\theta) \). Here, remembering that \( \sec^2(\theta) - 1 = \tan^2(\theta) \) makes the transition seamless. By doing so, we convert the square root of complex-looking expressions into something more tractable, in this case to \( 2 \tan(\theta) \). Consequently, this reduces the integral into a form that is easier to manage, often leading to functions that are straightforward to integrate, like \( \cos(\theta) \).

• \( \sec^2(\theta) = 1 + \tan^2(\theta) \)
• \( \tan^2(\theta) = \sec^2(\theta) - 1 \)

Constant of Integration

When solving an indefinite integral, the constant of integration, denoted by \( C \), is crucial. It represents the infinite set of antiderivatives, as integration is essentially the reverse process of differentiation. Every function has a derivative, but the original function is not unique unless a constant is specified. For instance, when evaluating \( \int \cos(\theta) \, d\theta = \sin(\theta) + C \), the \( + C \) affirms that different original functions could produce the same derivative. The constant ensures the completeness of the solution and needs to be included when reporting results of indefinite integrals.