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403
15 Integration 2 – Further Techniques
Function
Integral
f(x)
c
f1x2 dx
1
xn
n
1
ex
xn
ex
1
x
sin x
cos x
cos x
sin x
sec2 x
tan x
ln x
2
cosec x
sec x tan x
cosec x cot x
1
cot x
sec x
cosec x
sin
21 x2
1
2a
2
sin
2
x
1
21
x2
1
2a
2
2
x
1
x2
1
1
2
1
x
1x
a
cos
1
x
cos
1
x
a
tan
1
x
1
x
tan 1
a
a
ax
ln a
2
a
1
x
ax
Example
sin 2x dx
cos 2x
2
c
Example
1
2x
1
dx
1
ln 2x
2
1
c
In this chapter we will look at the techniques of integrating more complicated functions
and a wider range of functions.
fichier-pdf.fr/2014/06/07/ibhm-403-445/
07/06/2014
For example, Lagrange in his paper Reflections on the
resolution of algebraic equations said that he would return to this question, indicating
that he still hoped to solve it by radicals.
fichier-pdf.fr/2014/06/07/ibhm-086-107/
07/06/2014
Supporting example or detail—either your insight or a critic’s comment
paraphrased or directly quoted
Never start a sentence with a quote that you have not introduced.
fichier-pdf.fr/2017/03/11/researchpapers/
11/03/2017
Steinhaus told Banach of a mathematical problem that he was working on without
making much headway and after a few days Banach had the main idea for the
required counter-example, which led to Steinhaus and Banach writing a joint paper.
fichier-pdf.fr/2014/06/07/ibhm-337-372/
07/06/2014
a⫹b苷b⫹a
ab 苷 ba
共a ⫹ b兲 ⫹ c 苷 a ⫹ 共b ⫹ c兲
a共b ⫹ c兲 苷 ab ⫹ ac
共ab兲c 苷 a共bc兲
(Commutative Law)
(Associative Law)
(Distributive law)
In particular, putting a 苷 ⫺1 in the Distributive Law, we get
⫺共b ⫹ c兲 苷 共⫺1兲共b ⫹ c兲 苷 共⫺1兲b ⫹ 共⫺1兲c
and so
⫺共b ⫹ c兲 苷 ⫺b ⫺ c
EXAMPLE 1
(a) 共3xy兲共⫺4x兲 苷 3共⫺4兲x 2y 苷 ⫺12x 2y
(b) 2t共7x ⫹ 2tx ⫺ 11兲 苷 14tx ⫹ 4t 2x ⫺ 22t
(c) 4 ⫺ 3共x ⫺ 2兲 苷 4 ⫺ 3x ⫹ 6 苷 10 ⫺ 3x
If we use the Distributive Law three times, we get
共a ⫹ b兲共c ⫹ d兲 苷 共a ⫹ b兲c ⫹ 共a ⫹ b兲d 苷 ac ⫹ bc ⫹ ad ⫹ bd
This says that we multiply two factors by multiplying each term in one factor by each term
in the other factor and adding the products.
fichier-pdf.fr/2017/09/27/6e-reviewofalgebra/
27/09/2017
However, it was
Nine Chapters on the Mathematical Art, written during the Han Dynasty in China between
200 BC and 100 BC, which gave the first known example of matrix methods as set up
in the problem below.
fichier-pdf.fr/2014/06/07/ibhm-268-304/
07/06/2014
Translate the data from frequency tables into a pictorial
representation…
Graphs for qualitative variables (example)
When Apple Computer introduced the iMac computer in August
1998, the company wanted to learn whether the iMac was
expanding Apple’s market share.
fichier-pdf.fr/2015/10/12/lecture-3/
12/10/2015
The key to production
• Reservoir model definition (Why a reservoir model ?)
• Example of history matching from a reservoir model
• Specificity of reliable cellular models
• Dynamic simulation to reduce reserves uncertainties
• Simples rules modeling for optimal
Cellular models
• Example of geological model
• Example of field
• Image parameters an hydrocarbon field
• Definition of a cellular model
Objectives
• Objectives of your company
• The reservoir engineer’s job
• Integrated reservoir studies:
fichier-pdf.fr/2012/05/07/microsoft-powerpoint-1-rcm-introduction-final/
07/05/2012