Exercises: Immediate integrals

Calculate the following indefinite integral of a basic polynomial function using the properties of linearity (separating sums and taking out constants):

Find the family of antiderivatives of the following function containing a square root. Remember to express the root as a fractional power before integrating:

To integrate the following rational function, you must first simplify the expression by dividing each term in the numerator by the denominator:

Calculate the indefinite integral of this function that combines an exponential with base $e$ and a negative power. Be careful with the sign when applying the power rule:

Find the antiderivative of the function $f(x) = \frac{1}{x} + \cos(x)$. Remember which function has a derivative of $\frac{1}{x}$ and pay attention to the modulus:

A market study indicates that the growth rate of followers for a new influencer is given by the function $f(x) = 6x + 10$ (where $x$ is the time in days).

Find the function $F(x)$ that gives the total number of followers, knowing that at the time of launch (day $x=0$) they already had $500$ followers.

Determine the value of the following near-immediate integrals (logarithmic and exponential types with constants):

1. a) $\int \frac{3}{x+2} \, dx$

1. b) $\int e^{5x} \, dx$

In a mobile phone case manufacturing company, the marginal cost (the cost of producing an additional unit) is estimated using the function $C'(x) = 0,6x + 4$.

Obtain the expression for total cost $C(x)$ if it is known that the company's fixed costs (rent, electricity, etc.), which correspond to producing 0 units, amount to $1200$ euros.

In an Applied Mathematics to Social Sciences II exam, Carlos has to solve the following indefinite integral:

To calculate it, he writes the following in his working:

1. a) Identify and explain in detail the conceptual error that Carlos made. Justify why the integral of a quotient is not the quotient of the integrals, relying on the relationship between differentiation and integration.
2. b) Explain how to algebraically rewrite the integrand, dividing each term by the monomial in the denominator, in order to apply the table of standard integrals.
3. c) Correct the exercise and find the correct family of antiderivatives.

To find the family of antiderivatives of the function $f(x) = (2x - 3)(x + 5)$, two classmates discuss which approach to take:

• Elena claims that, since it is a product of two factors that depend on $x$, it is mandatory to use integration by parts taking $u = 2x - 3$ and $dv = (x + 5) \, dx$.

• Sergio suggests first expanding the product using the distributive property to obtain a polynomial and then integrating term by term using the power rule for standard integrals.

1. a) Reasonably compare both proposals: explain which of the two methods is the most direct and appropriate for this problem, and whether Elena's proposal is wrong or simply unnecessarily complex.
2. b) Solve the integral following the most efficient strategy and check your result by differentiating the obtained function.

Calculate the indefinite integral of the following polynomial function containing terms with fractional and negative powers. Remember to express the result in the most simplified form possible:

Find the antiderivative function $F(x)$ of the function $f(x) = \frac{3x^2 + 2}{x^3 + 2x - 5}$ knowing that its graph passes through the point $P(1, 4)$. First analyse whether the numerator is the derivative of the denominator.

A courier company has determined that the rate of change of its monthly maintenance costs (in euros) is given by the marginal cost function $C'(x) = 0,09x^2 - 0,8x + 20$, where $x$ represents the number of vans in the fleet. If fixed costs (the cost when there are no vans) amount to $1500$ euros, obtain the total cost function $C(x)$.

Calculate the integral of the following exponential function by adjusting the necessary constant so that the integrand matches the derivative of the composite function:

Determine the indefinite integral of the trigonometric function given below. Note that it is necessary to make a small numerical adjustment to obtain an immediate integral:

Find a function $f(x)$ whose derivative is $f'(x) = \frac{1}{\sqrt{3x + 1}}$ and which satisfies the condition $f(1) = 0$.

Solve the following indefinite integral of a rational function that results in an arctangent-type function. Remember that you must manipulate the denominator so that it has the form $1 + [g(x)]^2$:

Calculate the indefinite integral of the following function using the properties of powers and the linearity of integrals:

An agricultural cooperative in Jaén dedicated to bottling extra virgin olive oil calculates that the marginal bottling cost (in hundreds of euros per thousand liters) follows the function:

where $x$ is the bottled volume in thousands of liters ($x \ge 1$). It is known that the initial fixed setup cost of the machinery before processing the first thousand liters ($x = 1$) is $12$ hundred euros, that is, $C(1) = 12$.

1. a) Model the total cost function $C(x)$ by determining the corresponding constant of integration using standard integrals.
2. b) Justify why, in the antiderivative of the term $-\frac{1}{x}$, it is not strictly necessary to write the absolute value in this specific case, taking into account the context and the domain of the variable $x$.
3. c) Calculate the projected total cost for a batch of $x = 4$ thousand liters (use the approximation $\ln 4 \approx 1,39$).

The Directorate-General for Traffic (DGT) monitors the rate of vehicles entering the access roads to Madrid via the A-6 motorway during a period of the return operation. Between minutes $t = 1$ and $t = 9$, the inflow rate (in tens of vehicles per minute) is given by the function:

We want to calculate the cumulative total of vehicles $N(t)$ entering the road starting from minute $t = 1$, such that $N(1) = 0$.

1. a) Before integrating, make a preliminary estimate of the total number of vehicles that will enter during those $8$ minutes: evaluate the rate at the endpoints $r(1)$ and $r(9)$, calculate the arithmetic mean of both rates, and multiply it by the length of the interval ($\Delta t = 8$ minutes).
2. b) Find the function $N(t)$ by decomposing the quotient into a sum of powers and integrating directly using the condition $N(1) = 0$.
3. c) Calculate the exact value of $N(9)$ and evaluate whether your preliminary estimate was reasonable, explaining why the linear average slightly overestimates or underestimates the actual value due to the concavity or convexity of the rate.

A well-known mobile app development company has determined that the growth rate of its daily revenue (in hundreds of euros) is given by the marginal revenue function $f'(x) = 3x^2 \cdot e^{x^3} + \frac{2x}{x^2+1}$, where $x$ represents the time in weeks since the release of its latest update. Knowing that at the time of release ($x=0$) the cumulative revenue was 500 euros ($5$ units in the function), determine the expression for the total revenue function $f(x)$.

Calculate the antiderivative of the function $g(x) = \frac{x^3 - 2x^2 + 5x - 3}{x^2 + 4}$ satisfying the condition that its graph passes through the point $P(0, 1)$. Hint: First perform polynomial division to decompose the fraction.

An influencer is analysing the rate of follower loss on a social network following a controversy. The function describing this change is $s'(t) = \frac{-6t}{(t^2+3)^2} - \cos(2t)$, where $t$ is time in days. If the influencer initially had $100.000$ followers, find the function $s(t)$ that allows calculating the number of followers at any time $t$.

Determine the family of functions $F(x)$ whose derivative is $f(x) = \frac{5}{1+9x^2} + \sqrt[3]{x^2}$. Of all of them, identify the one whose tangent line at the point with x-coordinate $x=0$ has the equation $y = 5x - 2$.

Find the indefinite integral of the function $h(x) = \frac{(e^x + 2)^2}{e^x} + \sin(x) \cdot \cos^4(x)$. To solve it, try to simplify the first term by expanding the square and splitting the fraction, and observe the chain rule pattern in the second term.

The marginal cost of production in an e-scooter factory is defined by the function $C'(q) = \frac{100}{2q+10} + 15$, where $q$ is the number of scooters manufactured. If the fixed costs of the factory (the cost when $q=0$) amount to $2.500$ euros, obtain the total cost function $C(q)$. How much would the total cost be if $50$ scooters are manufactured?

Calculate the function $f(x)$ knowing that its second derivative is $f''(x) = 12x^2 - 6x + 2$ and that its graph passes through the point $A(1, 10)$, where the slope of the tangent line equals $5$.

Calculate the following indefinite integral using the properties of powers and the rules for standard integrals:

Next, find the antiderivative that takes the value $4$ when $x=1$.

A financial analysis department determines that the function modeling the accumulated profit $B(x)$ of an investment fund (in thousands of euros, with $x > 0$) has a family of antiderivatives given by the expression:

1. a) Solve the inverse problem: determine the original marginal function $b(x) = B'(x)$ and express it as a single algebraic fraction of the form $\frac{P(x)}{x^2}$, where $P(x)$ is a polynomial in standard form. State the degree and the coefficients of $P(x)$.
2. b) Prove theoretically that no function of the form $\frac{Q(x)}{x^2}$, where $Q(x)$ is a polynomial, can yield a term containing the function $\arctan(x)$ upon integration. Base your reasoning on the table of derivatives and standard integrals and on the integer powers resulting from dividing by a monomial.
3. c) Knowing that after $x = 1$ year the accumulated profit is $B(1) = 20$ thousand euros, find the constant $C$ and reason whether the marginal function $b(x)$ has any inflection point in the interval $(0, +\infty)$ by analyzing the sign of its second derivative.

Red Eléctrica de España (REE) records the instantaneous rate of change of electricity consumption in an industrial estate during the early morning using the function $D'(t)$ (in megawatts per hour, MW/h), where $t \in [1, 6]$ represents the hours elapsed since midnight:

1. a) Algebraically decompose the quotient into powers of $t$ and calculate the general cumulative consumption function $D(t)$ using standard integrals.
2. b) The decomposition yields the term $-\frac{5}{t} = -5t^{-1}$. Formally justify why it is not possible to apply the power rule $\int t^n \, dt = \frac{t^{n+1}}{n+1}$ to this term: explain what mathematical problem arises if $n = -1$ is substituted, and what transcendental function resolves this indeterminacy and why.
3. c) If at one o'clock in the morning ($t = 1$) the meter recorded a cumulative consumption of $D(1) = 4e + \frac{5\pi}{2}$ megawatt-hours (MWh), calculate the exact value of the constant of integration $C$ by simplifying the symbolic expressions involving $e$ and $\pi$ (recall that $\arctan(1) = \frac{\pi}{4}$ rad), and write the final expression for $D(t)$.