Evaluation of Definite Integral Evaluation of Definite Integral 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 integralExample 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), Know More About Whole Number Addition
=> ∫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. Evaluation of Definite Integral by substitution Definite integral substitution provides a simple way to solve the integral problem. It is similar to the indefinite integral substitution but the difference is that we have to deal with the limits in this case. Recall the methods of evaluating definite integral by first evaluating the indefinite integral and putting range on it. However in this method, we have to first use the substitution rule to find the Learn More About Whole Number Subtraction
indefinite integral and our second step would be evaluation of the expression formed. We can do the evaluation process in two ways. The first way is very obvious that we can do it at this point and the second approach is to evaluate at the time when we are in trouble at any point. Let us better understand with the help of following examples: Example 1: Evaluate the given definite integrals from range -2 to 0. âˆŤ2t2âˆš1-4t3.dt Solution: We will start with the way of evaluation we described above. As we already said, we would get in trouble in this method if we will not careful as there is a point which leads to a wrong evaluation. Way 1: If we forget about the limits of the definite integral for some time, it would be easy as we are solving an indefinite integral. The substitution can be done as follows: v=1-4t3
By using fundamental theorem of calculus we can calculate definite integrals in terms of indefinite integrals, this process is shown below-...