Questions: Use the accompanying sections of a table of integrals to evaluate the following indefinite integral. The integral may require preliminary work, such as completing the square or changing variables, before it can be found in a table. ∫ dx / sqrt(144 x^2 - 169), x > 13/12 Click here to view basic integrals. Click here to view trigonometric integrals. Click here to view reduction formulas for trigonometric functions. Click here to view integrals involving squares of x and a. Click here to view integrals involving ax+1-b. Click here to view other integrals. ∫ dx / sqrt(144 x^2 - 169) = □

Solution Steps

To solve the integral \(\int \frac{d x}{\sqrt{144 x^{2}-169}}\), we can use a trigonometric substitution. Notice that the expression under the square root, \(144x^2 - 169\), resembles the form \(a^2 \sec^2(\theta) - a^2\), which suggests using a substitution involving the secant function. Specifically, we can let \(x = \frac{13}{12} \sec(\theta)\), which simplifies the expression under the square root to a form that can be integrated using standard trigonometric integrals.

We start with the integral

\[ \int \frac{d x}{\sqrt{144 x^{2}-169}}. \]

To simplify the expression under the square root, we can rewrite it as

\[ \sqrt{144 x^2 - 169} = \sqrt{(12x)^2 - (13)^2}. \]

This suggests a trigonometric substitution, specifically \(x = \frac{13}{12} \sec(\theta)\).

Using the substitution, we find that

\[ \int \frac{d x}{\sqrt{144 x^{2}-169}} = \frac{1}{12} \log(12x + \sqrt{144x^2 - 169}) + C, \]

where \(C\) is the constant of integration.

Final Answer