Exercises: Semi-immediate integrals

Calculate the following indefinite integral of a composite power function, making the necessary adjustment to the constant so that the integrand matches the derivative of the inner function:

Find the antiderivative of the exponential function whose derivative requires a small numerical adjustment to the coefficient:

Determine the indefinite integral of the following rational function. Observe that the numerator is almost the derivative of the denominator, so it is a logarithmic-type integral:

Calculate the integral of the following trigonometric function. Remember that when integrating $\cos(f(x))$, you must take into account the derivative of the argument $f'(x)$:

Solve the integral of the following root function. To make the calculation easier, express it first as a power with a fractional exponent:

In a study of an advertising campaign on social media, it has been determined that the growth rate of an account's followers, $x$ days after the start of the campaign, is given by the function $f'(x) = \frac{x}{x^2 + 9}$, with $x \geq 0$.

Calculate the general expression of the total followers function $f(x)$, knowing that it is a logarithmic-type integral:

Calculate the integral of the following power function multiplied by a constant. Make sure to identify correctly the derivative of the base to adjust the coefficient:

A courier company has calculated that its marginal cost (the cost of sending an additional package) follows the function $C'(x) = 6 \cdot e^{3x}$. Find the expression for the total cost function $C(x)$ by calculating its indefinite integral:

In a written test, Elena had to compute the following indefinite integral:

Her working was as follows:

1. a) Differentiate the function $F(x) = 5 \ln|3x + 2|$ obtained by Elena and check whether it matches the original integrand. Explain in your own words what numerical factor she forgot to include and why the chain rule requires dividing by 3 when integrating expressions with a linear argument $ax+b$.
2. b) Write the correct step-by-step solution and determine the specific antiderivative $G(x)$ whose graph passes through the point $(0, 2)$.

You are asked to compute the indefinite integral:

1. a) Method 1: First expand the square of the binomial and integrate the resulting polynomial term by term.
2. b) Method 2: Solve it directly as a semi-immediate power-type integral $\int [u(x)]^n \cdot u'(x) \, dx$, adjusting the necessary constant so that the derivative of the inner argument appears.
3. c) Algebraically expand the expression obtained in Method 2 and explain why both solutions are correct and equivalent despite differing in the numerical constant term. Which of the two methods would you choose if the problem asked you to compute $\int (4x - 1)^6 \, dx$? Justify your choice.

A content creator has analysed that the growth rate of their followers on a social network follows a marginal function given by $f'(x) = (2x + 5)^3$, where $x$ represents the time in months. Calculate the family of antiderivatives $F(x)$ describing the total number of followers.

A courier company determines that the marginal cost of sending a parcel to a distance $x$ (in hundreds of kilometres) is given by the function $C'(x) = \frac{2x + 1}{x^2 + x + 4}$. Determine the total cost function $C(x)$, knowing that there are fixed management costs (when $x=0$) of $10$ euros.

The charging rate of a graphene battery in a laboratory behaves according to the function $v(t) = t \cdot e^{t^2}$, where $t$ is time in minutes. Calculate the function representing the total accumulated energy $E(t)$, taking into account that at the initial time $t = 0$, the accumulated energy is $5$ units.

Calculate the indefinite integral of the following function, applying the rule for logarithmic-type functions:

The figure shows the graph of the integrand $f(x)=\dfrac{\cos(x)}{2+\sin(x)}$.

Find the antiderivative of the function $f(x) = \frac{x}{\sqrt{x^2 + 9}}$ whose graph passes through the point $P(4, 10)$. Hint: Express the root as a power with a negative exponent to identify the form of the composite function.

The marketing department of a trainer brand has detected that daily sales $V(t)$ vary seasonally according to the derivative $V'(t) = 100 \cdot \cos(2t)$, where $t$ is time in days since the start of the campaign. Find the total sales function $V(t)$ knowing that on the first day ($t=0$), $500$ sales were made.

Solve the following indefinite integral by identifying the structure of the derivative of a composite function (power type):

Given the function $f(x) = \frac{e^x}{1 + e^x}$, you are asked to:

1. a) Find its general antiderivative $F(x)$, written in the form $F(x) = \ldots + C$, where $C$ is the constant of integration.

1. b) Determine the value of the constant $C$ so that the antiderivative satisfies $F(0) = \ln(2) + 3$.

Consider the family of continuous rational functions depending on a real parameter $k \neq 0$:

1. a) Find the general family of antiderivatives $F_k(x) = \int f_k(x) \, dx$, indicating which constant you must multiply and divide by to transform it into a logarithmic-type integral $\frac{u'(x)}{u(x)}$.
2. b) Determine the exact values of $k$ and the constant of integration $C$ knowing that the antiderivative sought satisfies two conditions simultaneously: its graph intersects the y-axis at $y = 1$ and it also passes through the point $(4, 1 + \ln(5))$.
3. c) Using the particular antiderivative obtained in the previous part, reason whether $F(x)$ has any points with a horizontal tangent in its domain without needing to apply differentiation rules from scratch.

In an intensive agricultural holding in Almería, the instantaneous flow rate of water demanded by a fertigation system during the first 4 hours of continuous operation is governed by the marginal function:

where $t$ represents the elapsed time in hours ($t \in [0, 4]$) and $c(t)$ is expressed in cubic metres per hour ($\text{m}^3/\text{h}$).

1. a) Before computing the antiderivative, make a reasoned estimate of the total volume of water consumed in those 4 hours: calculate the flow rate at the beginning ($t=0$) and at the end of the interval ($t=4$), and use these two extreme values to establish a lower bound and an upper bound for the total consumption.
2. b) Model the accumulated water volume function $V(t) = \int c(t) \, dt$ as a semi-immediate composite power-type integral with a linear argument, knowing that at the initial instant no water had been consumed yet ($V(0) = 0$).
3. c) Using $V(4)$, calculate the exact volume consumed after 4 hours and check whether your estimate from part a) was consistent with this actual value.

Calculate the antiderivative of the function $f(x) = \frac{x^2}{\sqrt{1 - x^6}}$ that passes through the origin of coordinates. The figure shows the graph of $f$, which is defined on $(-1,\,1)$ and has vertical asymptotes at $x=-1$ and $x=1$. Hint: Try to relate the exponent in the numerator to the one in the denominator to identify an arcsine-type structure.

A new open-world video game consumes mobile data at a transfer rate given by the function $D'(t) = \frac{e^t}{1 + e^{2t}}$, where $t$ is the time in hours and $D'(t)$ is measured in gigabytes per hour (GB/h). If at the start of playing ($t=0$) the data counter was at $0,5$ GB, determine the function $D(t)$ that allows calculating the cumulative total consumption at any time.

Determine the indefinite integral of the following function, making the necessary constant adjustments so that the numerator is the exact derivative of the inner function:

An influencer analyses the growth rate of their followers on a social network. The growth rate is given by $S'(t) = \frac{300 \cdot \ln t}{t}$ followers per day, where $t$ is time in days elapsed since the launch of an advertising campaign ($t \ge 1$). Given that at the initial time ($t=1$) they had $5.000$ followers, find the expression for the function $S(t)$.

Find all the antiderivatives of the function $f(x) = \frac{x + 2}{x^2 + 4x + 13}$.

Calculate the antiderivative $F(x)$ of the function $f(x) = \text{tg } x \cdot \sec^2 x$ satisfying the condition $F\left(\frac{\pi}{4}\right) = 2$. What value would the function take at $x = 0$?

In a biology laboratory, the spread of a bacterium is studied. The rate of change of the population $P(t)$ as a function of time $t$ (in hours) follows the law:

If initially ($t=0$) there are $1.000$ bacteria, determine the bacterial population after $2$ hours.

Find the function $f(x)$ knowing that its second derivative is $f''(x) = 12x^2 - 6x$ and that the tangent line to its graph at the point with x-coordinate $x = 1$ is the line $y = 4x - 2$.

Note: This exercise requires integrating twice consecutively functions with composite power structures.

According to statistical data from the Instituto Nacional de Estadística (INE) on incidents in metropolitan transport, the rate at which user complaints are registered (in hundreds of people per hour) during a working day between 6:00 ($t=0$) and 14:00 ($t=8$) is described by the marginal function:

where $t$ is the elapsed time in hours since 6:00.

1. a) Identify the semi-immediate structure of the integrand: express the numerator as a multiple of the derivative of the denominator, write the necessary constant adjustment, and find the family of antiderivatives $R(t) = \int r(t) \, dt$.
2. b) Determine the specific function $R(t)$ that expresses the accumulated number of complaints (in hundreds) knowing that at 6:00 in the morning no accumulated complaints were recorded ($R(0) = 0$).
3. c) A spokesperson for the transport company publicly states that before 14:00 ($t=8$) 1500 complaints will not be exceeded under any circumstances. Model and analyse analytically whether this statement is true or false. If it is false, calculate the exact time of day (in hours and minutes) when complaint number 1500 was reached.

In an exam, a student proposes the following solution for the indefinite integral $\int e^{-x^2} \, dx$:

"Since the derivative of the exponent is $(-x^2)' = -2x$, I multiply by $-2x$ inside the integral and compensate by dividing by $-2x$ outside it to have the derivative multiplying:

$$\int e^{-x^2} \, dx = -\frac{1}{2x} \int (-2x) e^{-x^2} \, dx = -\frac{1}{2x} e^{-x^2} + C$$"

1. a) Differentiate the function $F(x) = -\frac{1}{2x} e^{-x^2}$ obtained by the student using the product or quotient rule and formally show that the result does not match the integrand function $e^{-x^2}$.
2. b) Explain the underlying conceptual error: why is the linearity property $\int k \cdot f(x)\, dx = k \int f(x)\, dx$ only applicable when $k$ is a non-zero real constant, and why does it not allow pulling out expressions that depend on the variable of integration $x$?
3. c) Relying on your previous explanation, analyse the following three integrals. Rigorously justify which of them are indeed semi-immediate and solve them by adjusting the corresponding constant, and identify which one is structurally impossible to solve as semi-immediate: