108 lines
2.3 KiB
Plaintext
108 lines
2.3 KiB
Plaintext
#import "./template/template.typ": exercise, example, notes, theorem, definition, proof, lemma
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#show: notes.with(
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class: "EE1M1",
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lecture: "Integrations by parts and partial decomposition fractions",
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number: 6,
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date: "2025-11-21",
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)
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== Integrations by parts
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#theorem(title: "unbounded integral")[
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Suppose $f$ and $g$ are differentiable functions. Then
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$ integral f(x) g^(')(x) = f(x)g(x) - integral f^(')(x) g(x) d x $
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Using $g^(')(x) d x = d v$ and $f^(')(x) d x = d u$ we can also write:
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$ integral u d v = u v - integral v d u $
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]
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#theorem(title: "bounded integral")[
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Suppose $f$ and $g$ are differentiable functions. Then
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$ integral_a^b f(x) g^(')(x) d x = [f(x)g(x)]_a^b - integral_a^b f^(')(x)g(x) d x $
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]
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#example[
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Can the integral be solved using both the substitution role and integration
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by parts?
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$ integral 1/x ln(x) d x $
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#emph("Substitution rule")
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#emph("Integration by parts")
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]
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#example[
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$
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integral e^(sin(x)) sin(x) cos(x) d x \
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u = sin(x) \
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d u = cos(x) d x \
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integral e^u u d u \
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k^(') (x) = e^u \
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h(x) = u \
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integral e^u u d u = k(u)h(u) - integral k(u)h^(')(u) d u \
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integral e^u u d u = e^u u - e^u = e^(sin(x)) sin(x) - e^(sin(x)) + C
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$
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]
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#example[
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$
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integral e^(2 x) cos(x) d x = \
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$
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]
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#example[
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$
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integral arctan(x) d x = \
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g^(')(x) = 1, g(x) = x \
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$
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]
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#exercise[
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$ integral ln(3x) d x $
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]
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#exercise[
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$ integral_0^(-pi) e^(-x) sin(x) d x $
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]
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#exercise[
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$ integral ln(x) / x sin(ln(x)) d x $
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]
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== Partial decomposition fractions
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#theorem(title: "partial decomposition fractions")[
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Let $f(x) = P(x)/Q(x)$, with $P(x)$ and $Q(x)$ polynomials.
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The *degree* of a polynomial is its highest power.
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Suppose $"degree"(P(x)) < "degree"(Q(x))$.
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*Distinct linear factors* If $Q(x) = (x-a)(x-b)$ with $a eq.not b$ then
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$ P(x) / Q(x) = A / (x-a) + B / (x-b) $
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*Irreducible quadratic factor* If $Q(x) = (x-a)(x^2 + b x + c)$ with $b^2 - 4c < 0$.
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Then:
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$ P(x)/Q(x) = A/(x-a) + (B x + C)/(x^2 + b x + c) $
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*Repeated linear factor* If $Q(x) = (x-a)^2 (x-b)$ with $a eq.not b$ then
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$ P(x)/Q(x) = A/ (x-a) + B /(x-a)^2 + C/(x-b) $
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]
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$
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(x-a)(x-a)^2(x-b) = (x-a)^3(x-b) \
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(A(x-a)^2 + B(x-a))(x-b) + C(x-a)^3
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$
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$ A / (x-1) + B / (x+1) = A (x+1) + B(x-1) = 1 \
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B = -1/2 \
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A = 1/2
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$
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$ 1 / ((x-1)(x^2 - 2x + 3)) = $
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