notes/education/math/MATH1220 (calc II)/Integration with Trig Identities.md

The below integration makes use of the following trig identities:

1. The Pythagorean identity: \sin^2(x) + \cos^2(x) = 1
2. The derivative of sine: \frac{d}{dx}sin(x) = cos(x)
3. The derivative of cosine: \dfrac{d}{dx} \cos(x) = -\sin(x)
4. Half angle cosine identity: \cos^2(x) = \frac{1}{2}(1 + \cos(2x))
5. Half angle sine identity: \sin^2(x) = \frac{1}{2}(1 - \cos(2x))
6. tan^2(x) + 1 = sec^2(x)
7. \dfrac{d}{dx}(\tan(x)) = \sec^2(x) \Rightarrow \int \sec^2(x)dx = \tan(x) + C
8. \dfrac{d}{dx}(\sec x) = \sec(x)\tan(x) \Rightarrow \int\sec(x)\tan(x) dx = \sec(x) + C

Examples

Evaluate the integral \int\sin^5(x)dx

1. With trig identities, it's common to work backwards with u-sub, so we know that