The Fundamental Theorem of Calculus

Find the value of the derivative of the following integral function at the point $x = 2$:

$$F(x) = \int_{0}^{x} \sqrt{t^3 + 1} \, dt$$

Explain which mathematical theorem you are using to solve the problem without needing to calculate the antiderivative.

A video game programmer is designing the trajectory of a projectile. The speed of the object as a function of time $t$ is given by the expression $v(t) = \int_1^t (3x^2 - 4x) \, dx$.

Apply the Fundamental Theorem of Calculus to find the expression for the instantaneous acceleration $a(t)$, knowing that acceleration is the derivative of speed with respect to time ($a(t) = v'(t)$).

Calculate the derivative function $F'(x)$ of the following integral function using the Fundamental Theorem of Calculus and the chain rule:

$$F(x) = \int_{1}^{x^2} \frac{\ln(t)}{t + 1} \, dt$$

State the domain of definition of the function $F(x)$ so that the integral is well-defined.

Let the function $f(t) = \frac{\ln(t)}{t^2}$ be defined for $t > 0$. A new integral function $G(x)$ is defined as:

$$G(x) = \int_{x}^{x^2} f(t) \, dt$$

1. a) Applying the Fundamental Theorem of Calculus and the chain rule, obtain the expression for the derivative $G'(x)$.
2. b) Determine the intervals of increase and decrease of $G(x)$ in its domain.
3. c) Find the value of $x > 1$ for which the function $G(x)$ reaches a relative maximum.

Given the integral function $H(x) = \int_{0}^{x} \frac{t-1}{t^2+4} \, dt$:

1. a) Find the intervals of increase and decrease of $H(x)$.
2. b) Determine the coordinates of the relative extrema (maxima or minima) of $H(x)$ and classify them.

Let the function be defined by the integral $F(x) = \int_{0}^{x} \frac{t - 1}{e^t + 1} \, dt$.

1. a) Find the critical points of $F(x)$ and classify them using the first derivative test.
2. b) Study the curvature of $F(x)$ and determine if it has any points of inflection.
3. c) Is there any value of $x$ for which $F(x)$ is negative? Reason your answer based on the sign of the integrand and the limits of integration.

Let the function $G(x)$ be defined by:

$$G(x) = \int_{0}^{x} (t^2 - 4)e^{t} \, dt$$

Find the equation of the tangent line to the graph of $G(x)$ at the point with x-coordinate $x = 0$.

Note: Remember that to find the y-coordinate of the point of tangency you must calculate $G(0)$ and for the slope you must apply the Fundamental Theorem of Calculus.

A team of engineers is designing the trajectory of a racing drone. The vertical speed of the drone in the time interval $[0, 4]$ seconds is given by the function $v(t) = \int_{0}^{t} \frac{1}{1 + z^4} \, dz$.

1. a) Without calculating the integral explicitly, justify why the drone is always rising or maintaining the same height.
2. b) Calculate the vertical acceleration of the drone at the instant $t = 1$ second.
3. c) At what instant is the vertical acceleration maximum? Justify your answer by studying the second derivative of the position.

Consider the integral function $L(x) = \int_{1}^{x} \frac{\sqrt{1 + t^3}}{t} \, dt$ defined for $x \geq 1$.

1. a) Show that $L(x)$ is a strictly increasing function in its domain.
2. b) It is known that the graph of $L(x)$ has a tangent line at the point with x-coordinate $x = 2$ whose equation is $y = mx + n$. Calculate the value of the slope $m$.
3. c) Determine the acceleration of the slope's growth (second derivative) at that same point.

The electricity consumption of a house during the first hour of the morning is modeled by the function $f(x) = e^x + x$, where $x$ is the time in hours. Calculate the total accumulated consumption, represented by the definite integral:

$$\int_{0}^{1} (e^x + x) \, dx$$

A solar thermal panel has a design whose upper cross-section is bounded by the function $f(x) = 3x^2 + 2$, where $x$ is measured in feet. Calculate the value of the definite integral in the interval $[0, 2]$ using Barrow's Rule.

Determine the value of the positive real parameter $a$ knowing that the area bounded by the curve $f(x) = 3x^2$, the $OX$-axis and the vertical lines $x = 0$ and $x = a$ is exactly equal to 27 square units. Use Barrow's Rule for this.

Given a continuous function $f(x)$ in the interval $[1, 8]$, the values of the following definite integrals are known:

• $\int_1^5 f(x) \, dx = 7$

• $\int_5^8 f(x) \, dx = -3$

Use the properties of the definite integral to calculate the value of $\int_1^8 2 \cdot f(x) \, dx$.

Without performing complex integration calculations, justify with reasoning why the value of the following integral is equal to $0$, based on the properties of odd functions and the symmetry of the interval:

$$\int_{-2}^{2} (x^5 + 3x^3 - x) \, dx$$

Let $f(x)$ be a continuous function in $\mathbb{R}$ such that $\int_{0}^{1} f(x) \, dx = 2$. Find the value of the constant $k$ so that the following equality holds, knowing that the area under the curve on the left has been shifted and scaled:

$$\int_{0}^{2} f\left(\frac{x}{2}\right) \, dx + \int_{1}^{2} x \cdot f(x^2 - 1) \, dx = k$$

Use the integration by substitution method in the definite integrals to relate them to the initial data.

Calculate the area of the plane region bounded by the graph of the function $f(x) = \sin(x)$, the x-axis $OX$ and the vertical lines $x = 0$ and $x = \pi$. Sketch the region before applying Barrow's Rule.

Determine the area of the region enclosed between the curve $f(x) = x^2 - 4$ and the $OX$-axis in the interval $[0, 2]$. Remember that if the function is negative in the interval, the area corresponds to the absolute value of the definite integral.

The speed of an electric vehicle prototype during the first 4 seconds of a test is given by the function $v(t) = t^2 - 3t + 4$ (in ft/s).

1. a) Calculate the total distance traveled by the vehicle between the instants $t = 1$ s and $t = 3$ s applying Barrow's Rule.
2. b) Justify whether the vehicle stops at any point during that interval.

Given the function $f(x) = |x^2 - 2x|$, calculate the value of the definite integral in the interval $[0, 3]$ using Barrow's Rule:

$$\int_0^3 |x^2 - 2x| \, dx$$

Briefly explain why it is necessary to split the interval of integration to solve this exercise.

Calculate the value of the definite integral of the following piecewise function in the interval $[0, 2]$:

$$f(x) = \begin{cases} e^{2x} & \text{if } 0 \le x \le 1 \\ \frac{e^2}{x} & \text{if } 1 < x \le 2 \end{cases}$$

Check beforehand if the function is continuous at $x = 1$ to ensure that it is integrable.

Given the function $f(x) = x^2 - 4$, consider the integral function $H(x) = \int_{0}^{x} |f(t)| \, dt$.

1. a) Express $H(x)$ as a piecewise function for values of $x \in [0, 5]$.
2. b) Study the differentiability of $H(x)$ at the point $x = 2$. Is the Fundamental Theorem of Calculus directly applicable at that point?
3. c) Use Barrow's Rule to calculate the exact value of $H(3)$.

The logo of a new streetwear brand is bounded by the graphs of the functions $f(x) = \operatorname{sen}(\pi x)$ and $g(x) = ax$, with $a > 0$, in the first quadrant.

1. a) Determine the value of $a$ so that the line $y = ax$ passes through the relative maximum of the function $f(x)$ in the interval $[0, 1]$.
2. b) For the value of $a$ found, calculate the area of the region enclosed between both graphs using Barrow's Rule.

Find the mean value of the function $f(x) = \sqrt{x+1}$ in the interval $[0, 3]$ and determine the point $c \in (0, 3)$ whose existence is guaranteed by the Mean Value Theorem for Definite Integrals. Interpret the result geometrically.

Calculate the value of the following limit using L'Hôpital's Rule and the Fundamental Theorem of Calculus:

$$\lim_{x \to 0} \frac{\int_{0}^{x^2} \left( e^{t^2} - 1 \right) \, dt}{x^6}$$

Explain each step with reasoning, justifying the differentiation of the integral function.