Let’s discuss the definite integral. Let us discuss the definite integral. Now listen carefully. This is the indefinite integral of a function, okay. Now what we do is we integrate this function from one initial value to one final value. This is called definite integral of f(x), definite integral of f(x) with respect to x from, this is known as the lower limit and this is known as the upper limit. So this kind of terminology and this kind of expression is basically indicate the definite integral of a function from lower limit x = a to upper limit x = b. Done? And what it will be? f(x), listen class carefully, why we first go for the indefinite integral because the solution of definite integral can be found out only by indefinite integral, listen carefully.

What is the first step in solving an indefinite integral? That is forget; about the lower limit and upper limit and that is a standard indefinite integral, f(x) dx. And with the help of standard formulas or with the help of standard rules, integrate that function and you will get the new function, that is if you integrate the sine x, you will get what, -cos x. So you will integrate f(x) in an very ordinary manner, in the same manner as you do with indefinite. Done.

Then we will draw a line over here and we will write x = a, the lower limit and  x = b, the upper limit, and then finally the final answer of your definite integral will be the value of new function at upper limit, value of your new function at upper limit minus value of new function at lower limit. This is called definite integral of a function f(x), from lower limit to upper limit. Done class? Now note one thing, the answer which we will get here would be a definite value or a definite function. That won’t be a familial function.

Observe in this answer there is no “+c” at the end of the value. Now we will just understand why did we name indefinite integral as indefinite integral, because we do not get any particular or any definite function from those operations. All we used to get is familial function that is there can be infinite number of possible answers. Instead of indefinite integrals, if we go for definite integral process, then we get as answer a definite value or a definite function; it’s not an indefinite value, it’s either a fixed value or a fixed function. There is no “+c” that is why it is named as definite integral.

So for this particular expression, relax take it easy, listen attentively, now you have to think in just one manner, we don’t have to bother about how or from where did this came about, if it is written in this manner, so how do we read it? We will first understand how to read it. How will you read out the expression stated in front of you? Sir it is the definite integral of function f(x) from lower limit x = a to upper limit x = b with respect to x. how to solve this? Step number one, forget about the limits and solve the general indefinite integral. You will get a new function, done.

Take one example. Don’t write now. We have to integrate x² dx from a to b. you forgot? What is the integral of x² dx? We will apply the standard formula or standard rule.