# Fundamental Theorems of Calculus

Questions on the two fundamental theorems of
Calculus are presented. These questions have been designed to help you **better understand and use these theorems**. In order to answer the questions below, you might first need to review these theorems.

## Questions with Solutions

### Question 1

**True or False**. The second fundamental theorem of calculus states that if

\[ F(x) = \int_{a}^{x} f(t) \, dt \]

then \( F '(x) = f(x) \).

__Answer :__True.

### Question 2

**True or False**. If \[ F(x) = \int_{-2}^{3x} \sin(t) \, dt \] then the second fundamental theorem of calculus can be used to evaluate \( F '(x) \) as follows

__Answer :__False.

Note that the upper limit in the integral above is \( 3x \) and not \( x \), hence the integral above has the form \[ F(x) = \int_{-2}^{u(x)} f(t) \, dt \] Using the chain rule, we can write \[ F '(x) = \dfrac{dF}{du} \cdot \dfrac{du}{dx} = 3 \sin (3x) \]

### Question 3

**True or False**. Using the first fundamental of calculus \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] we can evaluate the following integral as follows \[ \int_{-1}^{1} \dfrac{1}{x^2} \, dx = -2 \]

__Answer :__False.

The interval of integration \( [-1 , 1] \) contains 0 at which function \( \dfrac{1}{x^2} \) is discontinuous and the above theorem cannot be applied.