Here are practice questions on the two fundamental theorems of Calculus[cite: 1]. These exercises are designed to help you better understand and apply these theorems. Each solution is hidden inside a collapsible dropdown so you can practice independently before reviewing the steps. Reviewing the theorems beforehand may be helpful.
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.
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 \[ F'(x) = \sin(3x). \]
Answer: False.
Note that the upper limit in the integral is \(3x\), not \(x\), so the integral has the form
\[ F(x) = \int_{-2}^{u(x)} f(t) \, dt. \]Using the chain rule, we get
\[ F'(x) = \frac{dF}{du} \cdot \frac{du}{dx} = 3 \sin(3x). \]True or False: Using the first fundamental theorem of calculus \[ \int_a^b f(x) \, dx = F(b) - F(a), \] we can evaluate \[ \int_{-1}^{1} \frac{1}{x^2} \, dx = -2. \]
Answer: False.
The interval of integration \([-1, 1]\) contains 0, where \( \frac{1}{x^2} \) is discontinuous. Therefore, the theorem cannot be applied in this case.
Additional questions with solutions[cite: 1] and tutorials and problems[cite: 1].