Problem 21 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{d x}{\left(x^{2}-1\right)^{3 / 2}}, \quad x>1 $$

Step-by-Step Solution

Answer

\(-\frac{1}{\sqrt{x^2-1}} + C\)

Step 1: Identify the Type of Integral

The given integral is \[ \int \frac{dx}{(x^2-1)^{3/2}}. \]This can be solved using a trigonometric substitution method because of its form involving an expression under the square root.

Step 2: Choose a Trigonometric Substitution

For the expression \( x^2 - 1 \), we use the substitution \( x = \sec(\theta) \). In this substitution, \( dx = \sec(\theta)\tan(\theta)\,d\theta \). Substituting these into the integral gives:\[\int \frac{\sec(\theta)\tan(\theta)\,d\theta}{(\sec^2(\theta) - 1)^{3/2}}.\]

Step 3: Simplify the Integral

Using the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \), the expression becomes:\[ \int \frac{\sec(\theta)\tan(\theta)\,d\theta}{(\tan^2(\theta))^{3/2}} = \int \frac{\sec(\theta)\tan(\theta)}{\tan^3(\theta)}\,d\theta. \]Simplifying, we have:\[ \int \frac{\sec(\theta)}{\tan^2(\theta)}\,d\theta = \int \frac{\sec(\theta)}{\sec^2(\theta) - 1}\,d\theta. \]

Step 4: Simplify Further and Integrate

Since \( \tan^2(\theta) = \sec^2(\theta) - 1 \), use substitution: \( u = \tan(\theta) \), \( du = \sec^2(\theta)\,d\theta \). Rewrite the integral in terms of \( u \):\[ \int \frac{1}{u^2} \cdot \frac{\sec^2(\theta)\,d\theta}{\sec^2(\theta)} = \int \frac{1}{u^2} \, du. \]This simplifies to:\[ \int u^{-2} \, du = -u^{-1} + C. \]

Step 5: Substitute Back to x

Since \( u = \tan(\theta) \), we have the expression:\[ -\frac{1}{\tan(\theta)} + C. \]Recall \( x = \sec(\theta) \), so \( \tan(\theta) = \sqrt{x^2-1} \). The answer in terms of \( x \) is:\[ -\frac{1}{\sqrt{x^2-1}} + C. \]

Key Concepts

Integral Calculus

Integral calculus deals with the determination of integrals and their properties.
It is a key area of calculus that lets us quantify areas under curves and solve physical problems involving accumulation.

• Definite Integral: Represents the accumulation of quantities, such as areas or distances over an interval.
• Indefinite Integral: Gives the antiderivative of a function, which includes a constant of integration.

In our exercise, we aim to evaluate an indefinite integral, \(\int \frac{dx}{(x^2-1)^{3/2}}.\)The expression under the integral can often be linked to trigonometric forms, which is why we explore special techniques, such as trigonometric substitution, to find solutions.
Understanding the structure of the function helps determine which techniques can simplify evaluation.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the parameters.
These identities are invaluable in converting complex expressions into simpler forms.
Some fundamental trigonometric identities include:

• Pythagorean Identities:
  • \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
  • \( 1 + \tan^2(\theta) = \sec^2(\theta) \)
• Reciprocal Identities:
  • \( \sin(\theta) = \frac{1}{\csc(\theta)} \)
  • \( \cos(\theta) = \frac{1}{\sec(\theta)} \)

In the context of our integral, we use the identity:\(\sec^2(\theta) - 1 = \tan^2(\theta),\)to simplify the expression under the square root and move forward with the calculation.

Substitution Method

The substitution method is a strategic approach used to simplify integrals by changing variables.
The goal is to transform a challenging integral into one that is easier to solve.
Here's how the substitution method typically works:

• Choose an appropriate substitution that simplifies the given integral. This often involves replacing parts of the expression with trigonometric functions.
• Determine the differential, and substitute both the variables and the differentials.
• Simplify the resulting expression and perform the integration.
• Finally, substitute back to the original variable to get the solution in terms of the initial variable.

In our specific problem, we used \( x = \sec(\theta) \) as a trig substitution because the expression \( x^2 - 1 \) can be represented by \( \tan(\theta) \) using the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \).
This transformation significantly simplifies the integral, allowing us to express it in more manageable terms and proceed with the integration.