4) \( \int \frac{d x}{5 x^{2}-75} \)

To tackle the integral \( \int \frac{d x}{5 x^{2}-75} \), we can first simplify the denominator. Factoring out the common term gives us \( 5(x^2 - 15) \). So the integral rewrites itself as: \[ \int \frac{d x}{5 (x^2 - 15)} = \frac{1}{5} \int \frac{d x}{x^2 - 15}. \] Now, we recognize that \( x^2 - 15 \) is a difference of squares, leading us to use the formula for integrating functions of the form \( \int \frac{dx}{x^2 - a^2}\). In this case, we can use the arctangent function, leading us to: \[ \frac{1}{5} \cdot \frac{1}{\sqrt{15}} \ln \left| \frac{x - \sqrt{15}}{x + \sqrt{15}} \right| + C, \] noting that the constant can be added for completeness. Therefore, the integrated result leads to an elegant expression embracing logarithmic properties. Enjoy integrating!