57 lines
3.3 KiB
Markdown
57 lines
3.3 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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$$-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x) $$
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# Trigonometric Substitutions
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Trigonometric substitution is useful for equations containing $\sqrt{a^2 + x^2}$ or $a^2 + x^2$, where $a$ is any constant. It removes any addition or subtraction.
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The general process involves the use of a trig identity and multiplying everything in that identity by a constant.
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Consider the identity:
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$$ 1 + \tan^2(\theta) = \sec^2(\theta)$$
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Multiplying both sides of the identity by $a^2$, we get:
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$$a^2 + a^2\tan^2(\theta) = a^2\sec^2(\theta)$$
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This enables us to make use of **substitution** to simplify many integrals.
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- $x = a\tan \theta$
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- $dx = a \sec^2\theta d\theta$
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- for $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$
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# Examples
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> Evaluate the integral $\int\frac{3}{4+x^2}dx$
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1. Move the constant coefficient out of the integral:
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$$ \int \frac{3}{4+x^2}dx = 3\int \frac{1}{4 + x^2}dx$$
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2. Let $x = 2tan\theta$ and $dx = (2sec^2\theta d\theta)$, substitute accordingly
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$$ = 3\int\frac{1}{4 + 4\tan^2\theta}(2\sec^2\theta)d\theta$$
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3. Factor $4$ out in the denominator
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$$ = 3\int\frac{1}{4(1 + \tan^2\theta)}(2\sec^2\theta)d\theta$$
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4. Considering the identity $1 + \tan^2 \theta = \sec^2 \theta$
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$$ = 3\int\frac{1}{4(\sec^2\theta)}(2\sec^2\theta)d\theta$$
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5. $\sec^2\theta$ is present in the numerator and the denominator, so we can cancel those out. This means that:
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$$ 3\int\frac{2}{4}d\theta = \frac{3}{2} \theta + C$$
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6. At this point, we want to determine what $\theta$ is equal to relative to $x$.
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1. Look back to step 2 we defined $x = 2\tan\theta$
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2. Moving $2$ to the other side, we get $\frac{x}{2} = \tan\theta$
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3. Because we defined bounds for our definition of $\theta$, we can take advantage of $\arctan$, therefore:
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$$ \theta = \arctan(\frac{x}{2}) $$
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7. Rewriting the equation with $\theta$ in terms of x, we get:
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$$ \frac{3}{2}\arctan(\frac{x}{2}) + C$$
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This means that:
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$$ \int\frac{3}{4+x^2}dx = \frac{3}{2}\arctan(\frac{x}{2}) + C $$ |