Exercises: Properties of the definite integral

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:

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

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.

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

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

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.

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 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.

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:

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

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: $\int_{0}^{5} v(t) \, dt$.

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:

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

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

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.

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.

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.

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 $\int_{-5}^{0} f(x) \, dx$.

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.

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 $I$.

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.

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 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 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.

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.