Title: Theory / Important / Math / Analysis / Zorich{ / 05 Primitive/ / 1 Single variable
Tags: integralprimitive récupéré

Name: A primitive of \(f\)
Definition: A function \(F:\mathbb{R\to R}\) such that \(F'=f\)
Page: 306
Chapter: 5
Section: 7.1
Variables: \(f:\mathbb{R\to R}\)
Notation: [$]\int f(x)\dd x[/$]
Name2: An indefinite integral of \(f\)

Question: What can be said of \(F\) and \(F'\) ?
Answer: Their difference is a constant
Page: 306
Chapter: 5
Section: 7.1
Variables: \(f:\mathbb{R\to R}\)
\(F,F'\) two primitives of \(f\)
Index: 1
Kind: Prop

Question: [$]\int\sum a_i f_i(x)\dd x[/$]
Answer: equals [$]\sum a_i\int f_i(x)\dd x[/$]
Page: 308
Chapter: 5
Section: 7.2
Variables: \(f_i:I\to \mathbb{R}\),
Index: a

Question: [$]\int (\prod f(x))'\dd x[/$]
Answer: equals [$]\sum\int f'_i f_{1-i}(x)\dd x+C [/$]
Page: 308
Chapter: 5
Section: 7.2
Variables: \(f_i:I\to \mathbb{R}\),
Index: b

Smaller0: [$]\defFun{[a,b]}{\mathbb R}{y}{\int_c^y(f\circ \phi)(t)\phi'(t)\dd t}[/$]
Greater0: [$]\defFun{[a,b]}{\mathbb R}{y}{((\int f\dd t)\circ\phi)(y)+C}[/$]
Page: 308
Chapter: 5
Section: 7.2
Variables: \(f:[c,d]\to \mathbb{R}\),
\(\phi:[a,b]\to [c,d]\)
Index: c
Smaller to greater: =
Assuming: \(\phi\) smooth mapping
Construction: Differentiate both side.

Name: The cosine integral
Definition: [$]\defFun{\mathbb R_{\ge0}}{\mathbb R}{x}{\int_x^\infty\frac{\cos x}{x}\dd x}[/$]
Page: 314
Chapter: 5
Section: 7.2c
Context: Analysis
Notation: \(\text{Ci}\)