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