## 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

### Properties of definite integral

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.

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.

### Summation of series by integration

### Gamma function

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

### Reduction formulae for definite integration

### Walli’s formula

### Leibnitz’s rule

(1) If f(x) is continuous and u(x), v(x) are differentiable functions in the interval [a, b], then,

(2) If the function and are defined on [a, b] and differentiable at a point and is continuous, then,

### Some important results of definite integral

### 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 x_{1}, x_{2}, x_{3}, ………. x_{n} in (*a*, *b*), then we can define subintervals (a, x_{1}), (x_{1}, x_{2}) ……… (x_{n-1}, x_{n}), (x_{n}, 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.