# 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**