Initial commit

new_notes/Calculus_Lecture_6.typ

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

new_notes/template/template.typ

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+#import "@preview/chic-hdr:0.5.0": *
+#import "@preview/theorion:0.4.1": *
+#import cosmos.fancy: *
+
+#let notes(
+  class : "",
+  lecture: "",
+  number: none,
+  date: none,
+  body
+) = {
+
+  show: chic.with(
+    chic-footer(
+      left-side: date,
+      right-side: chic-page-number()
+    ),
+    chic-header(
+      left-side: class,
+      right-side: [Lecture #number : #lecture]),
+    chic-separator(1pt)
+  )
+
+  show heading: set text(fill: black)
+  show heading.where(level: 1): set heading(numbering: none)
+  set heading(numbering: (first, ..nums) => numbering("1", ..nums))
+  show heading.where(level: 2): set align(center)
+  show heading.where(level: 2): set text(size: 18pt, font: "New Computer Modern")
+
+  show heading: it => {
+    if it.level == 2 {
+      block[
+        #smallcaps[Chapter. ]
+        #counter(heading).display(it.numbering)
+        #smallcaps[#it.body]
+      ]
+    } else {
+      block[#it.body]
+    }
+  }
+
+  show: show-theorion
+
+  set-primary-body-color(red.lighten(80%))
+  set-primary-border-color(red.darken(50%))
+
+  set-secondary-border-color(green.darken(50%))
+  set-secondary-body-color(green.lighten(80%))
+  set-secondary-symbol[#sym.suit.diamond.filled]
+
+  [= Lecture #number: #lecture]
+
+  outline(title: none, target: figure.where(kind: "theorem"))
+
+  outline(title: none, target: figure.where(kind: "definition"))
+
+  line(length: 100%, stroke: 1pt)
+
+  body
+}