Lecture III
Indefinite integral. Definite integral
Lecture questions
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
CxFdxxf )()(
• 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:
• The differential of the indefinite integral is equal to integration element:
• The general antiderivative of a constant times a function is the constant multiplied by the general antiderivative of the function (The constant multiple rule):
• If f(x) and g(x) are defined on the same interval, then:
)()( xfdxxf
dxxfdxxfd )()(
dxxfkdxxkf )()(
dxxgdxxudxxfdxxgxuxf )()()())()()((
Formulas of integrating of some functions
Cxdx
Cn
xdxx
nn
1
1
Cxdxx
ln1
Ca
adxa
xx
ln
Cedxe xx
Cxxdx sincos
Cxxdx cossin
Ctgxdxx2cos
1
Cctgxdxx2sin
1
Techniques of integration
• Method of direct integration using integral formulas and properties
• Integration by substitution
• Integration by Parts
duxvxvxudvxu )()()()(
Definite integral
Curvilinear trapezoidThe 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
n
iii xCfS
1
)(
nniin xCfxCfxCfxCfS )(...)(...)()( 2211
Definite integral
n
iii
xn
b
a
xCfdxxfi
10max
)(lim)(
• 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.
Fundamental Theorem of Integral Calculus.Newton – Leibniz formula
b
a
b
a
xFaFbFdxxf )()()()(
• If f(x) is continuous and F(x) is any arbitrary primitive for f(x) i.e. any function such that
then)()( xfxF
Thank you for your attention !
