Problem 54 - Calculus Early Transcendentals

Question

Evaluate the following definite $$\int_{10 / \sqrt{3}}^{10} \frac{d y}{\sqrt{y^{2}-25}}$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral: $$\int_{\frac{10}{\sqrt{3}}}^{10}\frac{d y}{\sqrt{y^2-25}}$$ Answer: $$\cosh^{-1}(2) - \cosh^{-1}(\frac{2\sqrt{3}}{3})$$

Step 1: Rewrite the integrand using hyperbolic function substitution

Let's rewrite the integrand, \(\frac{1}{\sqrt{y^2 - 25}}\), using a hyperbolic function substitution. Let \(y = 5\cosh{(\theta)}\), then \(dy = 5\sinh{(\theta)}d\theta\). Note that \(\cosh^2{(\theta)} - 1 = \sinh^2{(\theta)}\). Therefore, the integrand becomes: $$\frac{5\sinh{(\theta)}d\theta}{\sqrt{25\sinh^2{(\theta)}}} = \frac{5\sinh{(\theta)}d\theta}{5\sinh{(\theta)}}$$

Step 2: Simplify the integrand and substitute the limits

Now, simplify the integrand and substitute the limits in terms of \(\theta\): $$\int_{\frac{10}{\sqrt{3}}}^{10}\frac{d y}{\sqrt{y^2-25}} = \int_{\cosh^{-1}(\frac{2\sqrt{3}}{3})}^{\cosh^{-1}(2)} 1 \thinspace d\theta$$

Step 3: Evaluate the definite integral

The integral is simplified, so we can evaluate the definite integral: $$\int_{\cosh^{-1}(\frac{2\sqrt{3}}{3})}^{\cosh^{-1}(2)} 1 \thinspace d\theta = \left[\theta\right]_{\cosh^{-1}(\frac{2\sqrt{3}}{3})}^{\cosh^{-1}(2)} = \left(\cosh^{-1}(2) - \cosh^{-1}(\frac{2\sqrt{3}}{3})\right)$$ The evaluated definite integral is: $$\int_{10 / \sqrt{3}}^{10} \frac{dy}{\sqrt{y^{2}-25}} = \cosh^{-1}(2) - \cosh^{-1}(\frac{2\sqrt{3}}{3})$$

Key Concepts

Hyperbolic Function Substitution

Hyperbolic function substitution is a technique used to simplify integrals involving expressions like \( \sqrt{y^2 - a^2} \). In our exercise, we have \( \sqrt{y^2 - 25} \), which suggests using a hyperbolic function substitution. To handle such expressions, we use the identity related to hyperbolic functions:

• \( \cosh^2{(\theta)} - 1 = \sinh^2{(\theta)} \)

The substitution \( y = 5\cosh{(\theta)} \) is chosen because it matches our expression structure where \( a = 5 \). This substitution transforms \( dy \) to \( 5\sinh{(\theta)}d\theta \). The initial integral becomes:\[\int \frac{5\sinh{(\theta)}d\theta}{5\sinh{(\theta)}}\]which simplifies to \( \int 1 \, d\theta \). This simplification is achieved because the terms \( \sinh{(\theta)} \) in the numerator and denominator cancel out.

Integral Evaluation

After simplifying the integrand, we are left with a simple form for the integral: \[\int 1 \, d\theta\]This expression indicates that the integration needs to account for the entire given range of \( \theta \). This results in finding the difference between the values of \( \theta \) at the limits available after substitution. Since the integral of 1 with respect to any variable is the variable itself, the solution involves evaluating:\[\left[ \theta \right]_{\text{lower limit}}^{\text{upper limit}}\]Performing this step is straightforward—subtract the value at the lower limit from the value at the upper limit of integration.

Definite Integral Limits

Once we've simplified the integral using hyperbolic substitution, it's important to express the original limits of the definite integral in terms of \( \theta \). Given the original limits are from \( y = \frac{10}{\sqrt{3}} \) to \( y = 10 \), we substitute them based on our substitution \( y = 5\cosh{(\theta)} \). So, for these values:

• Calculate \( \theta = \cosh^{-1}\left(\frac{y}{5}\right) \), which provides the required limits.

Using this method:

• For \( y = \frac{10}{\sqrt{3}} \), \( \theta = \cosh^{-1}\left(\frac{2\sqrt{3}}{3}\right) \)
• For \( y = 10 \), \( \theta = \cosh^{-1}(2) \)

This transformation allows the definite integral to be properly evaluated. Inserting these limits into the integral provides the final expression from the indefinite integral evaluation, confirming the bounds of \( \theta \).