A Definite Integral is an integral with limits. The definite integral is of the form ∫ ab f( x ) dx where a, b, and x can be complex Numbers. The definite integral can be also be defined as- Let f( x ) be a continuous function on [ p , q ] and let F ( x ) is Antiderivative of f ( x ) then ∫pq f( x ) dx = F( x )|pq = F( p ) - F( q ).
By using fundamental theorem of Calculus we can calculate definite integrals in terms of indefinite integrals, this process is shown below-
∫ab f( x ) dx = F( b ) - F( a ),
here F is the definite indefinite integral for function f( x).
Let us take some examples understand how to evaluate definite integral-
Example 1) Calculate I( a ) = ∫0∏/2 dx / 1 + ( tan x )a.
Solution) As we know tan ( ∏ / 2 – x ) = cot x.
Let z = (tan x )a ,
So I( a ) = ∫0 ∏ / 4 dx / (1 + z) + ∫∏/4 ∏/2 dx / (1 + z),
=> ∫0∏/4 dx / ( 1 + z ) + ∫0∏/4 dx / (1 + 1 / z),
=> ∫0∏/4 ( 1 / (1 + z) + 1 / (1 + 1 / z)) dx,
=> ∫0∏/4 dx,
=> 1 / 4 ∏.
So from above example we learnt that evaluating definite integrals involves the process shown below-
First we have to find the indefinite integral, then we will find the Functions that are not continuous at any Point between the limits of Integration. Also note that the function should be continuous in the interval of integration. This is how we evaluate the definite integral.