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education/math/MATH1220 (calc II)/Integration with Trig Identities.md
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@ -7,7 +7,11 @@ The below integration makes use of the following trig identities:
6. $tan^2(x) + 1 = sec^2(x)$ 6. $tan^2(x) + 1 = sec^2(x)$
7. $\dfrac{d}{dx}(\tan(x)) = \sec^2(x) \Rightarrow \int \sec^2(x)dx = \tan(x) + C$ 7. $\dfrac{d}{dx}(\tan(x)) = \sec^2(x) \Rightarrow \int \sec^2(x)dx = \tan(x) + C$
8. $\dfrac{d}{dx}(\sec x) = \sec(x)\tan(x) \Rightarrow \int\sec(x)\tan(x) dx = \sec(x) + C$ 8. $\dfrac{d}{dx}(\sec x) = \sec(x)\tan(x) \Rightarrow \int\sec(x)\tan(x) dx = \sec(x) + C$
# Examples # Examples
> Evaluate the integral $\int\sin^5(x)dx$ > 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 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:
$$ \int(1)