Lecture III
Indefinite integral.
Definite integral
Lecture questions
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Antiderivative
Indefinite (primitive) integral
Indefinite integral properties
Formulas of integrating some functions
Curvilinear trapezoid. Area of a curvilinear trapezoid. Riemann Sum
Definite integral
Fundamental Theorem of Integral Calculus
Newton – Leibniz formula
Antiderivative. Indefinite integral
Antiderivative
•
Antidifferentiation (integration) is the
inverse operation of the differentiation.
• In calculus, an antiderivative of a function f(x) is
a function F(x) whose derivative is equal to f(x)
F ′(x) = f(x)
or
dF=f(x)dx
Antiderivative
• Any constant may be added to F(x) to get the
antiderivative of the function f(x).
• Antidifferentiation (or integration) is the
process of finding the set of all antiderivatives of
a given function f(x)
Antiderivative
The entire antiderivative
family of f(x) can be
obtained by changing the
value of C in F(x); where C
is an arbitrary constant
known as the constant of
integration. Essentially, the
graphs of antiderivatives of
a given function are vertical
translations of each other;
each
graph's
location
depending upon the value
of C.
Indefinite integral
 f ( x)dx F ( x)  C
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Terminology:
 - integral symbol
x – integration variable
f(x) - integrand (subintegral function)
f(x)dx - integrand (integration element)
C – constant of integration
Integral properties
• The first derivative of the indefinite integral is equal to
subintegral function:

 f (x)dx  f (x)
• The differential of the indefinite integral is equal to
integration element:
d  f ( x) dx  f ( x) dx
• The general antiderivative of a constant times a function
is the constant multiplied by the general antiderivative of
the function (The constant multiple rule):
 kf ( x)dx k  f ( x)dx
• If f(x) and g(x) are defined on the same interval, then:
 ( f ( x)  u ( x)  g ( x)) dx  f ( x)dx   u ( x)dx  g ( x)dx
Formulas of integrating of some
functions
 dx  x  C
n 1
x
n
x
 dx  n  1  C
1
 x dx  ln x  C
x
x
e
dx

e
C

ax
 a dx  ln a  C
x
 cos xdx  sin x  C
 sin xdx   cos x  C
1
 cos 2 x dx  tgx  C
1
 sin 2 x dx  ctgx  C
Techniques of integration
• Method of direct integration using integral
formulas and properties
• Integration by substitution
• Integration by Parts
 u ( x)dv  u ( x)v( x)   v( x)du
Definite integral
Curvilinear trapezoid
The figure, bounded by the graph of a function y=f(x),
the x-axis and straight lines x=a and x=b, is called a
curvilinear trapezoid.
Area of a curvilinear trapezoid.
Riemann Sum
Sn  f (C1 )x1  f (C2 )x2  ... f (Ci )xi  ... f (Cn )xn
n
S   f (Ci )xi
i 1
Definite integral
• The smaller the lengths Δxi of the subintervals, the
more exact is the above expression for the area of
the curvilinear trapezoid. In order to find the exact
value of the area S, it is necessary to find the limit
of the sums Sn as the number of intervals of
subdivisions increases without bound and the
largest of the lengths Δxi tends to zero.
b
 f ( x)dx 
a
n
lim
 f (C )x
n
max xi 0 i 1
i
i
Fundamental Theorem of Integral Calculus.
Newton – Leibniz formula
• If f(x) is continuous and F(x) is any arbitrary primitive
for f(x) i.e. any function such that
F ( x)  f ( x)
then
b
 f ( x)dx  F (b)  F (a)  F ( x)
a
b
a
Thank you for your attention !