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. In the above example, we can convert the equation into simpler cosine functions by setting du to -\sin(x)dx. This means that u is equal to cos(x).

\int\sin^4(x)\sin(x)dx

2. Rewrite sin^4(x) to be (\sin^2(x))^2 to take advantage of the trig identity 1 - \cos^2(x) = \sin^2(x)

\int(\sin^2x)^2 \sin(x)dx

3. Apply the above trig identity and substitute u:

\int(1 - u^2)^2 (-du)

4. Foil out and move negative out of integral:

-\int(1 - 2u^2 + u^4)du

5. Take advantage of the distributive property of integrals:

- (u - \frac{2}{3}u^3 + \frac{1}{5}u^5) + C

6. Substituting \cos(x) back in for u, we get the evaluated (but not entirely simplified) integral:

-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x)