Welcome back to class! Tonight, we are tackling Section 7-2: Trigonometric Integrals. In this section, we combine our knowledge of differentiation rules with trigonometric identities to integrate complex combinations of trig functions. While this topic can initially seem like a lot of memorization, it really comes down to recognizing specific patterns.

Here is an overview of the key strategies covered in the lecture notes attached below.

1. Powers of Sine and Cosine

When evaluating integrals in the form $ \int \sin^m x \cos^n x \, dx $, the strategy depends on whether the powers are odd or even.

  • If the power of Cosine is odd ($n = 2k + 1$):
    Save one factor of $\cos x$ to handle the $du$. Convert the remaining cosine factors to sine using the identity $\cos^2 x = 1 - \sin^2 x$. Then, substitute $u = \sin x$.
  • If the power of Sine is odd ($m = 2k + 1$):
    Save one factor of $\sin x$. Convert the remaining sine factors to cosine using $\sin^2 x = 1 - \cos^2 x$. Then, substitute $u = \cos x$.
  • If both powers are even:
    We cannot use simple $u$-substitution immediately. Instead, we use the half-angle identities to reduce the powers: $$ \sin^2 x = \frac{1}{2}(1 - \cos 2x) \quad \text{and} \quad \cos^2 x = \frac{1}{2}(1 + \cos 2x) $$

2. Powers of Tangent and Secant

For integrals involving $ \int \tan^m x \sec^n x \, dx $, we look for relationships between the derivatives of tangent (which is $\sec^2 x$) and secant (which is $\sec x \tan x$).

  • If the power of Secant is even ($n$ is even, $n \ge 2$):
    Save a factor of $\sec^2 x$. Convert the remaining secants to tangents using $\sec^2 x = 1 + \tan^2 x$. Then, substitute $u = \tan x$.
  • If the power of Tangent is odd ($m$ is odd):
    Save a factor of $\sec x \tan x$. Convert the remaining tangents to secants using $\tan^2 x = \sec^2 x - 1$. Then, substitute $u = \sec x$.

3. Product-to-Sum Identities

Finally, you may encounter integrals with different arguments, such as $ \int \sin(4x)\cos(5x) \, dx $. For these, we use product-to-sum identities to split the product into sums that are easy to integrate individually. For example:

$$ \sin A \cos B = \frac{1}{2}[\sin(A-B) + \sin(A+B)] $$

Key Takeaways

Success in this section requires patience and a good handle on your trig identities. Don't forget the standard integrals for tangent and secant, which often appear at the end of these problems:

  • $\int \tan x \, dx = \ln |\sec x| + C$
  • $\int \sec x \, dx = \ln |\sec x + \tan x| + C$

Be sure to download the full notes below to see step-by-step solutions for Examples 2 through 9.