# 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
Just replace t by x.

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

Now Consider Q.1 of GRE Mathematics (GR9768)

Solution:
By using Fundamental Theorem of Calculus:

We have,

Solution:

Now consider a little complex problem

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

Hence,

Q. Differentiate

Solution:

As we know that

Hence,