#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)) =  $