Mathematics → Subject 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.