Class notes 201-NYB-05

Antiderivatives


definition : F(x) is an antiderivative of f(x) on an interva
on I


F(x) (f(x) differenciation f(x) is the derivative
f (x) (F(x) antidifferenciation

General antiderivative
taken from mean value theorem (f'(x) g'(x) , then f(x) = g(x)
i
=



If f(x) & g(x) are both antiderivatives of a function h(x

Basic antiderivative rules
- if
you are asked to evaluate (f(x)dx ,
you are asked to
> f(x) = F'(x) for some function F(x)

· The basic antiderivative rules are the basic derivative rule




Indefinite
Integral
f f(x)dx =
F(x) + C
<
any two antiderivatives of a same function m
constant


Substitution rule
< chain rule backwards :
(f(g(x)) .


g'(x)dx =
f f(u)du , let u




< visual v=g(x) is usually inside something (ex expo
cues : :


<
U'=g'(X) is being multiplied by dx (on the same level
<sometimes work is necessary to find g(x) (ex laws of ex :


fractions polynomial long division etc
, h ,




recall :
Ix + a2dx =

&arctan(a) + C



Si 1ax=-02
arcsin( *) + C




Integration by parts :
product rule backwards


recall :
[f(x)g(x)) =
f'(x)g(x) +
g'(x) f(x)
So :

((f(x)g(x()'dx f(f'(x)g(x) g'(x)f(x))dx = +



<
f(x)g(x) f f'(x)g(x)dx (g'(x) f(x)dx
= +

,