1. The document discusses the history and concepts of integrals in mathematics.
2. Key figures mentioned include Bernhard Riemann, who developed the formal definition of integrals, and Henri Lebesgue, who developed the theory of integration.
3. Integral substitution and partial integration are introduced as methods for evaluating integrals. Examples are provided to demonstrate these methods.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
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3. I G H T
C A H A Y A
L
The Greatest strategy is dommed if it’s
implemented badly
Bernhard Riemann
Quote
Strategi terbesar akan hancur jika
diterapkan dengan buruk
Allah(pemberi) cahaya (kepada)
langit dan bumi. Perumpamaan
cahaya-Nya, seperti sebuah
lubang yang tidak tembus, yang
di dalamnya ada pelita besar.
Pelita itu di dalam tabung
kaca(dan) tabung kaca itu
bagaikan bintang yang berkilauan,
yang dinyalakan dengan minyak
dari pohon yang diberkahi, (yaitu)
pohon zaitun yang tumbuh tidak
di timur dan tidak pula di barat,
yang minyaknya (saja) hampir-
hampir menerangi, walaupun
tidak disentuh api. Cahaya di atas
cahaya(berlapis-lapis), Allah
memberi petunjuk kepada
cahaya-Nya bagi orang yang Dia
kehendaki, dan Allah membuat
perumpamaan-perumpamaan
bagi manusia. Dan Allah Maha
Mengetahui segala sesuatu
(Surah An-Nur Ayat 35)
4. Sejarah tentang integral
History
01
Pengantar dan pengertian integral
Introduction
02
Definisi dan Contoh Integral Fungsi
Integral Algebra Function
03
Latihan dan Aplikasi Integral
Exercises and Applied
04
The Name
of Game
5. History of Integral
Sejarah Integral
Eudoxus
Method of
Exhaustion
Hasan Ibn
al-Haytham
Menghitung volume
paraboloid
Leibniz, Newton
The Fundamental
Theorem of
Calculus
Lebesgue, Darboux,
Henstock-Kurzweil
Generalization of
Integration
Archimedes, Liu Hui
Menghitung area
lingkaran, luas permukan
dan volume
Cavalieri
Method of Invisibles
Riemann
The formal definition
of Integral
NOW
1854
17 M
16 M
1040 m
3 SM
370 SM
6. Excellent People
Orang Luar Biasa
Georg Friedrich
Bernhard Riemann
(1826-1866)
Matematikawan pertama
menemukan konsep formal
tentang Integral dengan limit
Henri Leon Lebesgue
(1875-1941)
Matematikawan yang terkenal
dengan “Theory of Integration”
Ralph Henstock
(1923-2007)
Integration theorist, dia bersama
Jaroslav Kurzweil(1957)
menemukan Integral Henstock-
Kurzweil.
7. Now People
Orang Jaman Now
Peng Yee Lee
(born 1938)
Studi tentang integral. Paper
“Lanzhou Lectures on
Henstock Integration”
Made Tantrawan
UGM, NUS
Paper tentang Integral
berjudul “Bounded Baire
Function and the Henstock-
Stieltjes Integral”
8. Introduction
Kenalan dulu, yuks
Definite Integral
Diketahui 𝑓 𝑥 = 𝑥 ⇒ ∫
!
"
𝑓 𝑥 𝑑𝑥
Derivative Function
Diketahui 𝑓 𝑥 = 𝑥 + 1 ⇒ 𝑓#
𝑥 = 1
Indefinite Integral
Diketahui 𝑓 𝑥 = 𝑥 ⇒ ∫ 𝑓 𝑥 𝑑𝑥
Anti Derivative Function
Diketahui 𝑓#
𝑥 = 1 ⇒ 𝑓 𝑥 =?
9. Integral of Properties
Sifat-sifat Integral
Integral Tak Tentu Fungsi Aljabar
* 𝑎𝑥$𝑑𝑥 =
𝑎
𝑛 + 1
𝑥$%" + 𝐶, 𝑛 ≠ −1
Sifat Penjumlahan
*
&
'
𝑓 𝑥 𝑑𝑥 = *
&
(
𝑓 𝑥 𝑑𝑥 + *
(
'
𝑓 𝑥 𝑑𝑥
Integral Nol
*
&
&
𝑓 𝑥 𝑑𝑥 = 0
Kelinearan
* 𝑘 𝑓 𝑥 𝑑𝑥 = 𝑘 * 𝑓 𝑥 𝑑𝑥
* 𝑓 𝑥 ± 𝑔 𝑥 𝑑𝑥 = * 𝑓 𝑥 𝑑𝑥 + *𝑔 𝑥 𝑑𝑥
Teorema Fundamental Kalkulus
Jika 𝑦 = 𝑓 𝑥 adalah fungsi yang kontinu
pada interval tertutup [𝑎, 𝑏], dan 𝐹(𝑥)
adalah sebarang antiturunan dari 𝑓(𝑥)
pada interval tersebut, maka berlaku
*
&
'
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 &
' = 𝐹 𝑏 − 𝐹(𝑎)
Sifat Kebalikan
Jika 𝑎 > 𝑏, maka dapat didefinisikan
*
&
'
𝑓 𝑥 𝑑𝑥 = − *
'
&
𝑓(𝑥) 𝑑𝑥
10. Method of Integration
Metode Pengintegralan
* 𝑓 𝑎𝑥 + 𝑏 𝑑𝑥 =
1
𝑎
𝑓(𝑎𝑥 + 𝑏)
Metode
Aljabar
* 𝑢 𝑑𝑣 = 𝑢𝑣 − *𝑣 𝑑𝑢
Metod
Parsial
Only at Indonesia, Really?
Metode Tanzalin merupakan shortcut dari
metode pengintegralan yaitu metode
Parsial. Metode banyak digunakan di
Indonesai. Namun beberapa investigasi
penelitian tentang metode ini tidak ada
benang merah antara nama Tanzalin.
Siapakah Tanzalin? Kenapa disebut
sebagai Tanzalin? Silakan di cek di mbah
Google. Banyak studi ini namun sampai
saat ini belum ditemukan asal muasal
metode Tanzalin.
Metode Tanzalin disebut juga
DI(Derivative-Integral atau Metode Tabular.
Metode
Tanzalin
*𝑓 𝑥 𝑔 𝑥 𝑑𝑥
dengan 𝑔 𝑥 = 𝑘 𝑓′(𝑥),
misalkan 𝑢 = 𝑓(𝑥)
Metode
Substitusi
11. Integral Substitusi
Integral substitusi adalah metode penyelesaian
masalah melalui integral dengan cara substitusi
kepada bentuk yang lebih sederhana, bentuk
sederhana yang dimaksud adalah berkaitan dengan
turunan suatu variabel.
Contoh Soal 1 :
! 𝑥!
− 1 𝑥 + 3 "
dx
Penyelesaian :
Misalkan jika 𝑢 = 𝑥 + 3 ⇒ 𝑑𝑥 = 𝑑𝑢 dan diperoleh 𝑥 = 𝑢 − 3, maka
! 𝑥! − 1 𝑥 + 3 "dx = ! 𝑢 − 3 ! − 1 𝑢"du = ! 𝑢! − 6𝑢 + 8 𝑢" du
! 𝑢#
− 6𝑢$
+ 8𝑢"
du =
1
8
𝑢%
−
6
7
𝑢#
+
4
3
𝑢$
+ 𝐶
Karena 𝑢 = 𝑥 + 3 sehingga didapat
! 𝑥! − 1 𝑥 + 3 "dx =
1
8
𝑥 + 3 % −
6
7
𝑥 + 3 # +
4
3
𝑥 + 3 $ + 𝐶
Misalkan ada bentuk integral
! 𝑓 𝑥 & 𝑔 𝑥 𝑑𝑥
maka dengan substitusi 𝑢 = 𝑓(𝑥) sehingga diperoleh
turunan dari 𝑢 adalah
𝑢' = 𝑓' 𝑥 =
𝑑𝑢
𝑑𝑥
⇒ 𝑑𝑥 =
𝑑𝑢
𝑓′(𝑥)
=
𝑑𝑢
𝑢′
didapat
! 𝑓 𝑥 &𝑔 𝑥 𝑑𝑥 = !𝑢& 𝑔 𝑥
𝑑𝑢
𝑢′
19. The Fundamental Theorem of Calculus
The Fundamental Theorem Calculus(FTC) atau
teorema dasar kalkulus, menetapkan hubungan
antara dua cabang kalkulus yaitu kalkulus diferensial
dan kalkulus integral.
Contoh Soal 1 :
𝑑
𝑑𝑥
!
(
-$
sec 𝑡 dt
Penyelesaian :
Menggunakan aturan rantai (Chain Rule), misalkan 𝑢 = 𝑥* maka diperoleh
𝑑
𝑑𝑥
!
(
-$
sec 𝑡 𝑑𝑡 =
𝑑
𝑑𝑥
!
(
6
sec 𝑡 𝑑𝑡 =
𝑑
𝑑𝑢
!
(
6
sec 𝑡 𝑑𝑡
𝑑𝑢
𝑑𝑥
= sec 𝑢
𝑑𝑢
𝑑𝑥
= sec 𝑥* . 4𝑥+
Teorema Dasar Kalkulus 1 ; Jika fungsi f kontinu pada
interval tertutup [a, b], maka fungsi 𝑔 didefinisikan oleh
∫
7
-
𝑓 𝑡 𝑑𝑡 𝑎 ≤ 𝑥 ≤ 𝑏
kontinu pada interval tertutup [𝑎, 𝑏] dan terdiferensial
pada interval terbuka (𝑎, 𝑏), dan nilai 𝑔' 𝑥 = 𝑓 𝑥 .
20. The Fundamental Theorem of Calculus
The Fundamental Theorem Calculus(FTC) 2 atau
teorema dasar kalkulus 2, metode yang lebih
sederhana dalam mencari nilai suatu integral.
Teorema ini mengacu pada integral tentu (definite
integral)
Contoh Soal 1 :
!
.!
(
𝑥+𝑑𝑥
Penyelesaian :
Diketahui fungsi 𝑓 𝑥 = 𝑥+
kontinu pada −2,1 dan mempunyai antiturunan 𝐹 𝑥 =
(
*
𝑥*
sehingga memberikan
hasil
!
.!
(
𝑥+𝑑𝑥 = 𝐹 1 − 𝐹 −2 =
1
4
1 * −
1
4
−2 * =
1
4
−
16
4
= −
15
4
Jadi, nilai
!
.!
(
𝑥+
𝑑𝑥 = −
15
4
Teorema Dasar Kalkulus 2 ; Jika fungsi f kontinu pada
interval tertutup [a, b], maka
∫
7
8
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)
dengan 𝐹 sebarang fungsi antiturunan 𝑓 mempunyai
sifat 𝐹'
= 𝑓.