91Ó°ÊÓ

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.

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.

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 \( \theta \).