Lesson date: March 3, 2025.

  • Use the Log Rule for Integration to integrate a rational function.
  • Integrate trigonometric functions.

Assignment

  • Vocabulary and teal boxes
  • pp353 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 43, 44, 47–49, 53, 55, 58, 68, 70, 71–77 odd 81, 92, 94, 104–107

Additional Resources

Log Rule for Integration

Despite the section’s length, it doesn’t cover anything too new. Instead, it combines the logarithm rule with integration by substitution.

\[\begin{align} \int \frac{1}{u} \, du = \ln|u| + C \end{align}\]

There is also the alternative version which can be helpful depending on the situation.

\[\begin{align} \int \frac{u'}{u} \, dx= \ln|u| + C \end{align}\]

This rule can help with integrating any function where the degree of the denominator is one more than the degree of the numerator, along with any situation where the numerator resembles the derivative of the denominator.

This section, like the one before on substitution, highlights what the book calls the “form-fitting” nature of integration. The typical first goal when working with an integral is to find a way to fit it into one of the forms you can work with. This is in contrast to differentiating, where you don’t need to worry about rewriting as an initial step.

There are a myriad of examples in the book, all of which you should go through. We’ll do a handful of them together in class.