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education/math/MATH1220 (calc II)/Integration with Trig Identities.md
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@ -4,7 +4,10 @@ The below integration makes use of the following trig identities:
3. The derivative of cosine: $\dfrac{d}{dx} \cos(x) = -\sin(x)$ 3. The derivative of cosine: $\dfrac{d}{dx} \cos(x) = -\sin(x)$
4. Half angle cosine identity: $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$ 4. Half angle cosine identity: $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$
5. Half angle sine identity: $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$ 5. Half angle sine identity: $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$
6. $tan^2(x) + 1 = sec^2(x) \Rightarrow \int \sec^2(x)$ 6. $tan^2(x) + 1 = sec^2(x)$
7. $\dfrac{d}{dx}(\tan(x)) = \sec^2(x)$ 7. $\dfrac{d}{dx}(\tan(x)) = \sec^2(x) \Rightarrow \int \sec^2(x)dx = \tan(x) + C$
8. 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