# Definite Integral

**Definite Integral**:

Definite integral is a form of Integral or Anti derivative in which we don’t get a range of answer or indefinite answer , Instead we get a fixed or definite answer.

Or, A definite integral is the integral of a function in a closed interval and it is denoted by:

Which means , the Integral or Anti derivative of the function “f(x)” in an interval from “a” to “b”.

**Definite Integral Formula**:

We can calculate or evaluate a definite integral using the definite integral formula which states that:

If “f” is continuous on [a,b] and is any antiderivative of “f” then ,

Now let us prove the formula:

We can prove the formula using the fundamental theorem of calculus which states that:

If “f” is continuous function and , then ,

Now let us prove the definite integral formula with the help of fundamental theorem of calculus,

Proof:

Let,

As “a” is a constant obviously we get “F(a) = 0″.

And as, “F” and are antiderivatives of same function “f” , they differ by a constant “c” as stated in antiderivatives .

So,

For some constant “c”.

Thus ,

or,

or ,

So ,

And we also have,

Thus ,

**Example of Definite Integrals**:

We can evaluate definite integrals using the same techniques of integration we used while evaluating indefinite integrals.

One example of definite integral is:

We can easily evaluate this integral using Integration Formulas .

So we get:

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