23 lines
1.4 KiB
Markdown
23 lines
1.4 KiB
Markdown
The below integration makes use of the following trig identities:
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1. The Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$
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2. The derivative of sine: $\frac{d}{dx}sin(x) = cos(x)$
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3. The derivative of cosine: $\dfrac{d}{dx} \cos(x) = -\sin(x)$
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4. Half angle cosine identity: $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$
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5. Half angle sine identity: $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$
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6. $tan^2(x) + 1 = sec^2(x)$
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7. $\dfrac{d}{dx}(\tan(x)) = \sec^2(x) \Rightarrow \int \sec^2(x)dx = \tan(x) + C$
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8. $\dfrac{d}{dx}(\sec x) = \sec(x)\tan(x) \Rightarrow \int\sec(x)\tan(x) dx = \sec(x) + C$
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# Examples
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> Evaluate the integral $\int\sin^5(x)dx$
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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 $cos(x)$.
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$$ \int\sin^4(x)\sin(x)dx$$
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2. Rewrite $sin^4(x)$ to be $(\sin^2(x))^2$ to take advantage of the trig identity $1 - \cos^2(x) = \sin^2(x)$
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$$ \int(\sin^2x)^2 \sin(x)dx$$
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3. Apply the above trig identity and substitute $u$:
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$$ \int(1 - u^2)^2 (-du) $$
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4. Foil out and move negative out of integral:
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$$ -\int(1 - 2u^2 + u^4)du $$
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5. Take advantage of the distributive property of integrals:
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$$ - (u - \frac{2}{3}u^3 + \frac{1}{5}u^5) + C $$
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6. Substituting $\cos(x)$ back in for $u$, we get the evaluated (but not entirely simplified) integral:
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$$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$ |