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
\( F '(x) = \sin (3x) \)
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.
