Properties of the definite integral

A drone travels horizontally at a constant velocity of $v(t) = 12$ meters per second. Calculate the distance traveled by the drone between seconds $t=5$ and $t=15$ by setting up the definite integral of the constant function:

$$\int_{5}^{15} 12 \, dt$$

Check the result using the geometric formula for the area of a rectangle.

Given two continuous functions $f(x)$ and $g(x)$ on the interval $[1, 5]$, we know the following values of their definite integrals:

$$\int_1^5 f(x) \, dx = 6 \quad \text{and} \quad \int_1^5 g(x) \, dx = -3$$

Use the linearity properties of the integral to calculate the value of:

$$\int_1^5 [2f(x) + 4g(x)] \, dx$$

It is known that an electrical signal received by a sensor follows a behavior described by two functions, $f(x)$ and $g(x)$. After conducting several measurements in the laboratory, the following values were obtained for their definite integrals over the interval $[1, 4]$:

$\int_{1}^{4} f(x) \, dx = 7$ and $\int_{1}^{4} g(x) \, dx = -3$

Using the linearity properties of the definite integral, calculate the value of the integral resulting from the combination of both signals in that interval:

$$\int_{1}^{4} [2f(x) - 5g(x) + 3] \, dx$$

Given a continuous function $h(x)$, the following data are known:

$$\int_1^3 h(x) \, dx = 4 \quad \text{and} \quad \int_1^6 h(x) \, dx = 10$$

Calculate the value of the integral $\int_3^6 h(x) \, dx$ using the property of the sum of intervals.

A logistics company monitors the speed of one of its vehicles. The graph of the function $f(x)$ represents the rate of fuel consumption. Knowing that:

$$\int_0^4 f(x) \, dx = 12 \quad \text{and} \quad \int_4^7 f(x) \, dx = 5$$

Calculate, applying the interval additivity property, the value of the integral $\int_0^7 f(x) \, dx$ and interpret its meaning if the x-axis represents the time in hours.

A surveillance drone travels along a straight path whose velocity as a function of time is given by a continuous function $v(t)$. Data analysts have determined that the accumulated displacement over different sections is:

1. From the start to second 3: $\int_{0}^{3} v(t) \, dt = 45$ meters.

2. From second 3 to second 8: $\int_{3}^{8} v(t) \, dt = 120$ meters.

3. From second 5 to second 8: $\int_{5}^{8} v(t) \, dt = 70$ meters.

Calculate, justifying the use of the interval additivity property, the displacement of the drone between seconds 0 and 5, that is: $\int_{0}^{5} v(t) \, dt$.

If it is known that the integral of a function $f(x)$ over a reversed interval is $\int_8^3 f(x) \, dx = 14$, determine the value of the integral in the conventional direction:

$$\int_3^8 f(x) \, dx$$

Justify which property of the definite integral you have used.

Determine, without needing to calculate the antiderivative or evaluate the function, the value of the following integral:

$$\int_{\sqrt{2}}^{\sqrt{2}} \frac{e^x \times \ln(x^2+1)}{\cos^2(x)+5} \, dx$$

State the name of the property on which you base your answer.

It is known that a function $f(x)$ is even and that the area enclosed under its curve between the y-axis ($x=0$) and the line $x=3$ is $4.5$ square units. That is:

$$\int_0^3 f(x) \, dx = 4.5$$

Calculate the value of $\int_{-3}^3 f(x) \, dx$, justifying the result using the symmetry of the function.

Consider a function $f(x)$ that is even and a function $g(x)$ that is odd across the set of real numbers. The following data are known:

$\int_{0}^{3} f(x) \, dx = 5$ and $\int_{-3}^{0} g(x) \, dx = 4$

Find, by applying the symmetry properties of the definite integral, the value of:

$$\int_{-3}^{3} [3f(x) - g(x)] \, dx$$

Without calculating the antiderivative, reason based on the symmetry properties of functions why the value of the following definite integral is exactly zero:

$$\int_{-10}^{10} (x^7 - \sin(x) + x^3) \, dx$$

In digital audio signal processing, functions that represent complex sound waves are studied. Consider the function $f(x) = x^3 \times \cos(x^2) \times e^{\sin^2(x)}$. Without computing any antiderivative, justify your reasoning based on the properties of the definite integral to find the value of:

$$\int_{-5}^{5} f(x) \, dx$$

Then, if it is known that $\int_{0}^{5} f(x) \, dx = K$, determine in terms of $K$ the value of $\int_{-5}^{0} f(x) \, dx$.

Without needing to calculate the antiderivatives, justify with reasoning which of the following integrals is greater using the integral comparison property ($f(x) \leq g(x) \implies \int f \leq \int g$):

$$I_1 = \int_{0}^{1} x^2 \, dx \quad \text{o} \quad I_2 = \int_{0}^{1} x^3 \, dx$$

Would your answer change if the interval of integration were $[1, 2]$? Justify your answer by analysing the behavior of the functions on both intervals.

Let $f(x)$ be a continuous function on the interval $[2, 6]$ whose minimum value is known to be $m = 1$ and whose maximum value is $M = 5$.

1. a) State, giving reasons, between which values the integral $\int_{2}^{6} f(x) \, dx$ lies.

1. b) If we also know that $\int_{2}^{6} f(x) \, dx = 12$, find the mean value of the function on that interval and explain its geometric meaning.

Given two continuous functions $f$ and $g$ on the interval $[0, 1]$ such that $f(x) = \frac{1}{1+x^4}$ and $g(x) = 1 - x^4$, use the boundedness property of the definite integral to prove the following inequality without calculating the integrals:

$$\int_{0}^{1} (1 - x^4) \, dx < \int_{0}^{1} \frac{1}{1+x^4} \, dx < 1$$

Hint: Analyse the behavior of the functions inside the interval $(0, 1)$ and use the fact that for $0 < u < 1$, $1-u < \frac{1}{1+u} < 1$ holds.

Knowing that the integral of a function $f(x)$ on the interval $[a, b]$ is equal to $I$, prove using the properties of the definite integral that:

$$\int_{a}^{b} [f(x) - \frac{I}{b-a}] \, dx = 0$$

Explain the interpretation of this result in relation to the Mean Value Theorem for Definite Integrals.

A surveillance drone moves in a straight line with a velocity given by the function $v(t) = t \times \sqrt{16 - t^2}$ in meters per second, where $t$ is the time in seconds from the start of the mission up to $t = 4$.

Find all times $c \in (0, 4)$ at which the instantaneous velocity of the drone equals its average velocity over the entire interval $[0, 4]$. To do this, apply the Mean Value Theorem for Definite Integrals and evaluate the corresponding definite integral.

Given the continuous function $f(x)$ defined on the interval $[-2, 5]$, it is known that the area enclosed by the curve and the $OX$ axis is as follows:

• On the interval $[-2, 1]$, the function is negative and the area of the region is $4$ $u^2$.

• On the interval $[1, 5]$, the function is positive and the area of the region is $10$ $u^2$.

Calculate, with reasoning, the values of the following definite integrals:

1. $\int_{-2}^{5} f(x) \, dx$

2. $\int_{5}^{1} f(x) \, dx$

3. $\int_{-2}^{5} |f(x)| \, dx$

Calculate the value of the definite integral of the function $f(x) = |x^2 - 4|$ on the interval $[0, 3]$. To do this, you must split the integral into several sections using the additivity property according to the sign of the integrand.

Let the function $f(x) = |x - a| + |x - 2|$ be defined on the interval $[0, 4]$. Given that the value of the definite integral of this function on that interval is a minimum for a specific value of $a \in [0, 4]$, use the properties of the integral to express $\int_{0}^{4} f(x) \, dx$ as a function of $a$, $I(a)$, and find the value of $a$ that minimises this integral.

A mechanical part has a top profile defined by the function $f(x)$, which is periodic with period $T=2$. It is known that in the interval $[0, 2]$, the function is $f(x) = x(2-x)$.

Calculate the total area under the curve in the interval $[0, 20]$ using the properties of the definite integral for periodic functions and additivity over intervals.

In an optics experiment on light diffraction, an integral of the form $I = \int_{0}^{\pi} \frac{x \times \sin(x)}{1 + \cos^2(x)} \, dx$ is obtained.

To solve it, use the reflection property $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a+b-x) \, dx$ to show that:

$$2I = \pi \times \int_{0}^{\pi} \frac{\sin(x)}{1 + \cos^2(x)} \, dx$$

From this equality, find the exact value of the integral $I$.

Let the function $F(x)$ be defined for $x > 0$ by the following integral with variable limits:

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

Using the Fundamental Theorem of Calculus and the chain rule, calculate the value of the derivative $F'(1)$. Explain which property of the definite integral allows you to split the interval of integration to apply the theorem when the limits are not constant.

Given the integral function $G(x) = \int_{0}^{x} (t^2 - 3t + 2) \times e^{-t^2} \, dt$, solve the following parts using the Fundamental Theorem of Calculus:

1. a) Calculate $G'(x)$, study its sign and analyze the intervals where $G(x)$ is increasing and decreasing in $[0, 3]$.

1. b) Determine the abscissas of the relative maxima and minima of $G(x)$ in the interval $[0, 3]$.

1. c) Without calculating the integral, justify whether the value of $\int_{1}^{2} (t^2 - 3t + 2) \times e^{-t^2} \, dt$ is positive or negative, based on the sign of the integrand in that interval.