Problem 44 - Calculus Single Variable

Question

Each of the integrands involves an expression of the form \(a^{2}-b^{2} x^{2}, a^{2}+b^{2} x^{2},\) or \(b^{2} x^{2}-a^{2} .\) Use an indirect substitution of the form \(x=(a / b) \sin (\theta), x=(a / b)\) \(\tan (\theta),\) or \(x=(a / b) \sec (\theta)\) to calculate the given integral. $$ \int_{4 / 5}^{1} \frac{\left(25 x^{2}-16\right)^{1 / 2}}{x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln\left(\frac{5}{4}\right) \).

Step 1: Identify the Substitution Form

The integrand contains the expression \(25x^{2} - 16\). To match this with the form \(b^2x^2 - a^2\), we recognize that \(a=4\) and \(b=5\). We will use the substitution \(x = \frac{4}{5} \sec(\theta)\).

Step 2: Perform the Substitution

When using the substitution \(x = \frac{4}{5} \sec(\theta)\), we differentiate to find \(dx = \frac{4}{5} \sec(\theta)\tan(\theta) d\theta\). Additionally, when \(x = \frac{4}{5}\), \(\theta = 0\); and when \(x = 1\), \(\theta = \cos^{-1}(\frac{4}{5})\).

Step 3: Simplify the Expression

Substitute \(x = \frac{4}{5} \sec(\theta)\) into the integral: \[\int_{0}^{\cos^{-1}(\frac{4}{5})} \left(\frac{4}{5} \right) \tan(\theta)\ d\theta\]. This makes use of the identity \(\tan(\theta) = \sqrt{\sec^2(\theta) - 1}\).

Step 4: Integrate

The integral reduces to: \[ \int_{0}^{\cos^{-1}(4/5)} \tan(\theta) d\theta \]This evaluates to \[ [-\ln|\cos\theta|]_{0}^{\cos^{-1}(4/5)} = -\ln(\frac{4}{5}) - (-\ln(1)) \].

Step 5: Calculate the Result

Substitute the boundaries into the resulting expression:\[= -\ln(\frac{4}{5})\]. Simplifying gives:\[\ln\left(\frac{5}{4}\right)\].

Key Concepts

Trigonometric Substitution

Trigonometric substitution is an integration technique often used when dealing with expressions involving quadratic polynomials within a radical. In our exercise, the integrand includes the expression \( 25x^2 - 16 \), which suggests using a trigonometric identity to simplify the integration process.
By recognizing this expression matches the form \( b^2x^2 - a^2 \), the substitution \( x = \frac{a}{b} \sec(\theta) \) is appropriate. Specifically, for \( a = 4 \) and \( b = 5 \), the substitution \( x = \frac{4}{5} \sec(\theta) \) is used.
This substitution helps transform the integral into a function involving trigonometric identities, which can simplify the computation of the definite integral. Understanding these relationships is key for handling integrals with quadratic expressions effectively.

Definite Integrals

Definite integrals are crucial in calculus for calculating the accumulated sum of continuous functions over a specific interval. In our problem, the definite integral \( \int_{4/5}^{1} \frac{\left(25x^2 - 16\right)^{1/2}}{x} \, dx \) represents a specific method to find the area under the curve from \( x = \frac{4}{5} \) to \( x = 1 \).
The process involves substituting the variable, adjusting the limits of integration according to the substitution, and evaluating the resulting simpler integral.
After transforming the integral with \( x = \frac{4}{5} \sec(\theta) \), the limits change to \( \theta = 0 \) and \( \theta = \cos^{-1}(\frac{4}{5}) \). The definite integral thus requires careful tracking and modifying of limits to ensure accurate solutions.

Calculus Problem-Solving

Problem-solving in calculus often involves breaking down complex problems into manageable steps. In this exercise, we initially identify a substitution to simplify integration.
The strategy involves:

• Identifying the general form of the expression to decide on the substitution.
• Applying trigonometric identities where applicable to transform the integrand.
• Computing the integral, adjusting for the substitution, and appropriately handling definite limits.

Each step builds on the previous to arrive at the simplest expression for integration. Understanding these logical steps is fundamental for tackling a wide variety of calculus problems effectively.

Integration Techniques

Various integration techniques serve as essential tools in calculus for solving integrals that are not straightforward. Among these techniques, trigonometric substitution is invaluable for integrals containing square roots of quadratic expressions.
Another critical technique highlighted in this exercise is the use of trigonometric identities such as \( \tan(\theta) = \sqrt{\sec^2(\theta) - 1} \), which aids in simplifying the integrand.
Mastering these techniques allows a smoother transition from a complicated-looking integral to an easily solvable form, ultimately making the calculus problem much simpler to resolve.