INTEGRALS YIELDING LOGARITHMIC FUNCTIONS \( \begin{array}{ll}\text { 1. } \int \frac{d x}{x-5} & \text { 2. } \int \frac{y d y}{5-y^{2}} \\ \text { 3. } \int \frac{d z}{z \ln z} 4 \cdot \int \frac{v^{2}+2 v+3 v}{v^{3}+3 v^{2}+9 v} d v \\ \text { 5. } \int_{1}^{2} \frac{1+2 x+x^{2}}{3 x+3 x^{2}+x^{3}} d x & 6 \cdot \int \frac{2 m^{2}+7 m-3}{m-2} d m 7 \cdot \int \frac{s^{3}-3 s^{2}+5}{s-3} d s \\ \text { College of Engineering } \\ \text { PHINMA Arailo University }\end{array} \)

To tackle integrals that yield logarithmic functions, the first step often involves recognizing that integrands of the form \( \frac{1}{x} \) or \( \frac{1}{ax + b} \) will lead directly to logarithmic results. For example, the integral \( \int \frac{dx}{x-5} \) simplifies to \( \ln |x-5| + C \). A friendly tip: always check for substitutions like \( u = x-5 \) to simplify the process! Another fun fact to consider is that when you're integrating rational functions that can be simplified, breaking them down with partial fractions makes the integration manageable! Take \( \int \frac{y \, dy}{5 - y^2} \): you can rewrite the denominator as a difference of squares and utilize properties of logarithms for a smooth integration. This can turn a potentially tricky integral into a stimulus for mathematical creativity!