Here’s a tip for next week’s recommended HOT topic standard:

3. Evaluate definite and indefinite integrals using trigonometric substitution.

To receive an H grade for your presentation, you will have to

  1. perform the appropriate trigonometric substitution,
  2. use trigonometric identities to simplify the integrand,
  3. use the appropriate integration technique to integrate,
  4. substitute back so your answer is in terms of the original variable,
  5. write your answer in simplest form.

WeBWorK *might* accept your answer even if it isn’t in simplest form, but for your presentation, your answer has to be in simplest form.

A good hint that your answer is not in simplest form is if you have the composition of a trigonometric function and an inverse trigonometric function. See the WeBWorK set Integration – Trigonometric Substitution: Problem 1. For this question, you are not integrating at all. Really, this exercise has you practicing just step 5 above.

Say, for example, we need to simplify $\cos(\sin^{-1}\left(\frac{x}{3}\right))$. It will be helpful to let $\theta = (\sin^{-1}\left(\frac{x}{3}\right))$, so that $\sin(\theta) = \frac{x}{3}$. This means that we need to simplify $\cos(\theta)$.

Now that we’ve rewritten the expression we need to simplify in this way, this is really just Warmup exercise 1 in Lesson 6 when $a=3$. See the solution on the Lesson 6 page (notice especially how the right triangle is labeled).