Fundamental Theorem of Calculus

MathematicsSubject Test → 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.
This 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. Answer.

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.