Fundamental Theorem of Calculus

Fundamental Theorem of Calculus Part-II. The Second Fundamental Theorem of Calculus.

Definition:

Let f be a continuous function on an interval I, and let a be any point in I. If F is defined by

then

at each point x in the interval I.
Which is the statement of the Second Fundamental Theorem of Calculus.
The above equation can also be written as

Fundamental Theorem of Calculus Part-I describes the relationship between integration and derivatives.

The upper limit is x, the lower limit is any constant. The variable of the function is other than x (Like t here). Hence, if differentiate with respect to x, the integral of a function in t, with upper limit x, then result is simply a function in x.

In other words this formula states:
"Where the integrand is continuous, the derivative of a definite integral with respect to its upper limit is equal to the integrand evaluated at the upper limit."

The following examples will make the matter more clear.

Applications of Fundamental Theorem of Calculus


Solution:
Apply Fundamental Theorem of Calculus to solve this question
Answer
Just replace t by x.


Solution:
Again apply Fundamental Theorem of Calculus to solve this question
Answer
Just replace t by x.

Now Consider Q.1 of GRE Mathematics (GR9768)

Solution:
By using Fundamental Theorem of Calculus:

We have,
Answer


Solution:

Answer

Now consider a little complex problem


For these types of questions where upper limit is not x, we have to remember,

Hence,

Answer

Q. Differentiate

Solution:

As we know that

Hence,
Answer

Q. Differentiate the following function

Solution:

Where a is any constant

Answer