Welcome back, class! In our lecture from February 7th, 2023, we dove deep into Section 7-3, covering two major techniques in our integration toolkit: handling powers of trigonometric functions and the powerful method of Trigonometric Substitution.

Calculus can get tricky when radicals appear in integrals, but these methods allow us to transform difficult algebraic expressions into manageable trigonometric ones. Let’s break down the key concepts from the class notes.

1. Integrals with Powers of Sine and Cosine

We started by looking at integrals in the form $\int \sin^m x \cos^n x \, dx$. As seen in the first example from the notes, when you have an odd power (like $\sin^3 x$), the strategy is to "peel off" one sine term to save for your $du$, and convert the remaining even powers using the identity $\sin^2 x = 1 - \cos^2 x$.

Example from class:
$$\int_0^{\pi/3} \sin^3 x \cos^5 x \, dx$$
By letting $u = \cos x$, this seemingly complex integral simplifies into a polynomial that is easy to evaluate!

2. Trigonometric Substitution

The core of this lesson focused on getting rid of square roots in the integrand. When you see expressions like $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$, or $\sqrt{x^2-a^2}$, we use specific substitutions to simplify the radical.

Here is the reference table we constructed in the notes (Page 7):

  • Form: $\sqrt{a^2 - x^2}$ → Sub: $x = a \sin \theta$ → Identity: $1 - \sin^2 \theta = \cos^2 \theta$
  • Form: $\sqrt{a^2 + x^2}$ → Sub: $x = a \tan \theta$ → Identity: $1 + \tan^2 \theta = \sec^2 \theta$
  • Form: $\sqrt{x^2 - a^2}$ → Sub: $x = a \sec \theta$ → Identity: \sec^2 \theta - 1 = \tan^2 \theta$

3. The Triangle Method (SOH CAH TOA)

One of the most important steps in Trigonometric Substitution is back-substitution. Once you integrate with respect to $\theta$, you must get your answer back in terms of $x$.

As demonstrated in the example $\int \frac{\sqrt{9-x^2}}{x^2} dx$, we arrive at an answer involving $\cot \theta$. To convert this back:

  1. Draw a right triangle.
  2. Use your original substitution (e.g., $\sin \theta = \frac{x}{3}$) to label two sides of the triangle.
  3. Use the Pythagorean theorem to find the third side (which usually looks like your original radical!).
  4. Read the required trig function directly from the triangle using SOH CAH TOA.

4. Check for Easier Methods First!

A crucial lesson from Page 13 of the notes involves the integral:

$$\int \frac{x}{\sqrt{x^2+4}} \, dx$$

While this looks like a candidate for Trig Substitution ($x = 2 \tan \theta$), notice that there is an $x$ in the numerator. This allows for a standard $u$-substitution where $u = x^2+4$ and $du = 2x \, dx$. Always check for a simple $u$-sub before diving into the longer Trig Sub process. It saves time and reduces algebraic errors!

5. Definite Integrals

Finally, we looked at definite integrals requiring changing the limits of integration. Remember, if you change variables from $x$ to $\theta$, you must also change your limits $a$ and $b$ to corresponding $\theta$ values. This often prevents the need to back-substitute at the very end.

Keep practicing these identities. Identifying which substitution to use is half the battle. You are doing great work—keep it up!