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Definite Integrals

December 10, 2020 by Prasanna

Definite Integrals

Let ?(x) be the primitive or anti-derivative of a function f(x) defined on [a, b] i.e., \(\frac { d }{ dx } [\phi (x)]=f(x)\). Then the definite integral of f(x) over [a,b] is denoted by \(\int _{ a }^{ b }{ f(x)dx }\) and is defined as [?(b) − ?(a)] i.e., \(\int _{ a }^{ b }{ f(x)dx } =\phi (b)-\phi (a)\). This is also called Newton Leibnitz formula.

The numbers a and b are called the limits of integration, ‘a’ is called the lower limit and ‘b’ the upper limit. The interval [a, b] is called the interval of integration. The interval [a, b] is also known as range of integration. Every definite integral has a unique value.

Evaluation of definite integral by substitution

When the variable in a definite integral is changed, the substitutions in terms of new variable should be effected at three places.
(i) In the integrand (ii)In the differential i.e., dx (iii) In the limits
Definite Integrals 1

Properties of definite integral

Definite Integrals 2
Generally this property is used when the integrand has two or more rules in the integration interval.
This is useful when is not continuous in [a, b] because we can break up the integral into several integrals at the points of discontinuity so that the function is continuous in the sub-intervals.
Definite Integrals 3
This property can be used only when lower limit is zero. It is generally used for those complicated integrals whose denominators are unchanged when x is replaced by (a – x).
Following integrals can be obtained with the help of above property.
Definite Integrals 4
Definite Integrals 5
Definite Integrals 6

Summation of series by integration

Definite Integrals 7

Gamma function

Definite Integrals 8
It is important to note that we multiply by (π/2); when both m and n are even.

Reduction formulae for definite integration

Definite Integrals 9

Walli’s formula

Definite Integrals 10

Leibnitz’s rule

(1) If f(x) is continuous and u(x), v(x) are differentiable functions in the interval [a, b], then,
Definite Integrals 11
(2) If the function and are defined on [a, b] and differentiable at a point and is continuous, then,
Definite Integrals 12

Some important results of definite integral

Definite Integrals 13
Definite Integrals 14

Integration of piecewise continuous functions

Any function f(x) which is discontinuous at finite number of points in an interval [a, b] can be made continuous in sub-intervals by breaking the intervals into these subintervals. If f(x) is discontinuous at points x1, x2, x3, ………. xn in (a, b), then we can define subintervals (a, x1), (x1, x2) ……… (xn-1, xn), (xn, b) such that f(x) is continuous in each of these subintervals. Such functions are called piecewise continuous functions. For integration of piecewise continuous function, we integrate f(x) in these sub-intervals and finally add all the values.

Filed Under: Mathematics Tagged With: Definite Integrals, Gamma function, Integration of piecewise continuous functions, Leibnitz’s rule, Properties of definite integral, Reduction formulae for definite integration, Some important results of definite integral, Summation of series by integration, Walli's formula

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