Exercises: Integration of rational functions

Calculate the following indefinite integral, observing that the numerator is, or can easily be transformed into, the derivative of the denominator:

A group of young engineers is monitoring the data flow of a server. The rate of growth of the stored data (in GB/min) is given by the function $v(t) = \frac{4}{t + 1}$, where $t$ is the time in minutes. Find the general expression for the data storage function $S(t)$.

Determine the indefinite integral of the following function, whose form corresponds to an arctangent-type antiderivative:

Decompose the integrand into partial fractions to calculate the integral of this rational function with simple real roots in the denominator:

Find the antiderivative of the function $g(x)$ by first adjusting the numerator to obtain an integer part and a fractional part:

In the study of alternating current electrical circuits, the current intensity $i(t)$ in amperes is usually modelled using functions combining exponential and trigonometric functions. Suppose the rate of change of the charge on a capacitor is given by the function:

Find the set of all antiderivatives $F(t)$ associated with this current using the method of integration by parts. Keep in mind that this type of integral is solved using the "iteration" or "cyclic" method.

Find the analytical expression of a function $f(x)$ such that its second derivative is $f''(x) = x \cdot \operatorname{sen}(x)$, knowing that the tangent line to the curve at the origin is the x-axis itself.

Find the function $f(x)$ that satisfies the condition of being continuous across the entire domain of definition of its branches, knowing that its derivative is given by:

and that $f(1) = 1$. You must integrate the second branch by parts and determine the value of the constants of integration to ensure continuity at $x = 0$.

An object moves along a straight line with a time-dependent acceleration defined by $a(t) = \ln(t+1)$. If at the initial moment ($t=0$) the object was at rest and at the position $s=10$ metres, determine the position equation $s(t)$ at any moment $t$.

Note: Remember that velocity is the integral of acceleration and position is the integral of velocity. Both integrals require the use of the integration by parts method.

Alba has to calculate the antiderivative $\int \frac{2x^2 - 3}{x^2 - 1} \, dx$. On her exam, she writes the following steps:

“Since the denominator decomposes into $(x-1)(x+1)$, I set up the decomposition directly:

Multiplying by the common denominator: $2x^2 - 3 = A(x+1) + B(x-1)$.

• If $x = 1 \implies -1 = 2A \implies A = -1/2$.

• If $x = -1 \implies -1 = -2B \implies B = 1/2$.

Therefore, the integral is $-\frac{1}{2}\ln|x-1| + \frac{1}{2}\ln|x+1| + C$”.

1. a) Without calculating the correct integral yet, explain in a reasoned manner why Alba's initial algebraic setup is flawed and what fundamental condition regarding the degrees of polynomials she has ignored.
2. b) Differentiate Alba's final result and verify that it does not match the original integrand, identifying which term is missing.
3. c) Solve the integral correctly by performing the preliminary polynomial division and the corresponding decomposition.

To solve the indefinite integral $I = \int \frac{2x + 3}{x^2 + 3x - 10} \, dx$, two classmates suggest different approaches:

• Marcos states: “The denominator has two distinct real roots, $x = 2$ and $x = -5$. Therefore, we must decompose the fraction into $\frac{A}{x-2} + \frac{B}{x+5}$, calculate $A$ and $B$, and integrate the sum of the two resulting logarithms”.

• Lucía replies: “That procedure is unnecessarily long for this specific case; the integral can be solved in a single line almost immediately without decomposing anything”.

1. a) Analyze both stances: are both mathematically valid? Justify which one is more efficient and what property or structure of the integrand allows Lucía's direct method to be applied.
2. b) Solve the integral using Lucía's method.
3. c) Slightly modify the numerator of the integral (keeping the same denominator $x^2 + 3x - 10$) so that Lucía's strategy is no longer directly applicable and it becomes essential to resort to Marcos's method of partial fractions. Then, solve the new integral you have proposed.

Use the integration by parts method to solve the following integral. Remember to choose $u$ and $dv$ appropriately following the mnemonic rule ALPES:

$\displaystyle \int x \cdot e^x dx$

Solve the following integral using the integration by parts method:

$\displaystyle \int (x + 2) \cdot \cos(x) dx$

Find the antiderivative of the natural logarithm function using the integration by parts method, taking $u = \ln(x)$ and $dv = dx$:

$\displaystyle \int \ln(x) dx$

A team of engineers is analysing the discharge rate of a new drone prototype's battery. They have determined that the function representing the rate of change of the charge (in mAh per minute) is given by $f(t) = 3t^2 - \frac{4}{t+1} + \sqrt{t}$, where $t$ is the time in minutes.

Find the total charge function $F(t)$, knowing that at the initial moment ($t = 0$) the drone's charge was $2500$ mAh.

An electronics components company estimates that its marginal cost (the cost of producing an additional unit) follows the function $C'(x) = x \cdot \ln(x)$, where $x$ is the number of thousands of units produced.

Determine the total cost function $C(x)$ knowing that the fixed costs (cost when $x = 1$, in this simplified model) are $10$ thousand euros.

Find the antiderivative of the function $g(x) = x^2 \cdot \cos(x)$ that passes through the origin. You will need to apply the integration by parts method twice in succession.

Find the indefinite integral $\int e^{2x} \cdot \sin(x) \, dx$.

This type of integral is called "cyclic". Apply the method of integration by parts until the original integral appears again and solve for its value.

Consider the rational function $f(x) = \dfrac{x + 4}{x^2 - 1}$, defined for $x \in [2, 4]$. The goal is to study the antiderivative $F(x) = \int f(x) \, dx$ that satisfies the condition $F(2) = 0$, and in particular to determine the value of $F(4)$.

1. a) Check that $f(x)$ is strictly decreasing and positive on the interval $[2, 4]$ by calculating the exact values of $f(2)$ and $f(4)$. From these values, make a reasoned estimation of the increment $F(4) - F(2)$ by bounding it between a minimum and a maximum value (hint: consider the area under a decreasing curve by comparing it with lower and upper rectangles constructed on the base of the interval $[2, 4]$).
2. b) Calculate the exact analytical expression of the antiderivative $F(x)$ using partial fraction decomposition and determine the value of the integration constant with the given condition.
3. c) Using the approximations $\ln 3 \approx 1,099$ and $\ln 5 \approx 1,609$, calculate the numerical value of $F(4)$ and verify whether your previous estimate from part a) was consistent and accurate.

Given the rational function $f(x) = \dfrac{a x^2 + b x + 5}{x^3 - x^2 - x + 1}$, where $a$ and $b$ are real coefficients:

1. a) Factor the denominator, indicating the roots and their multiplicity, and write the theoretical structure of its partial fraction decomposition.
2. b) Determine the values of the parameters $a$ and $b$ so that the following two conditions are met simultaneously:

1. In the family of antiderivatives of $f(x)$, the coefficient of the logarithmic term $\ln|x-1|$ is equal to zero.

2. The tangent line to the graph of any antiderivative $F(x)$ at the point with abscissa $x = 2$ has slope $m = 1$.

1. c) Using the values of $a$ and $b$ found in the previous section, obtain the expression for the specific antiderivative $F(x)$ whose graph passes through the point $P(2, 3)$.

Calculate the antiderivative of the function $f(x) = \dfrac{x^4 + x^3 - 2}{x^3 - x^2}$ that passes through the point $P(2, 5)$. Note that, due to the degrees of the polynomials, you must first perform polynomial division and then decompose the remaining fraction taking into account the presence of repeated real roots.

Determine the indefinite integral of the function $g(x) = \dfrac{1}{x^4 - 1}$. To solve this exercise, you must fully factorise the denominator over the real numbers and decompose the fraction into a sum containing terms with simple real roots and a term with complex roots yielding an arctangent form.

A team of engineers is monitoring the flow of coolant in a thermal power plant. The rate of change of the volume of liquid with respect to time $t$ is given by the function $v(t) = \dfrac{4t + 10}{t^2 + 4t + 13}$. Find the expression for the total volume $V(t)$ knowing that at the initial instant ($t=0$) the tank contained $100$ litres.

Note: Observe that the denominator has no real roots, so the solution will combine a natural logarithm and an arctangent.

We want to find a function $F(x)$ whose derivative is $f(x) = \dfrac{x^2 - 3x + 2}{(x+1)^3}$. Find this function knowing that its graph passes through the point $(0, 2)$. This problem requires the use of partial fractions with real roots of multiplicity three.

Calculate the indefinite integral $\displaystyle \int \frac{3x^2 + 1}{x^3 + x^2 + x + 1} \, dx$.

To do this, note that the denominator can be factorised using Ruffini's rule or by grouping terms, resulting in the product of a linear factor and an irreducible quadratic factor.

Determine the values of the parameters $A$, $B$ and $C$ so that the following equality holds and, subsequently, calculate the antiderivative of $f(x)$ that is zero at $x=0$:

A market study analyses the growth rate of sales for a new model of electric scooter. The function describing the rate of change of sales (in hundreds of units per month) is given by:

$f(x) = \dfrac{x^2 + 2x + 3}{x + 1}$

Where $x$ is the time in months. Calculate the family of total sales functions $F(x)$ by finding the antiderivative of this function. (Note: Perform polynomial division first).

Calculate the indefinite integral of the following rational function by decomposing it into partial fractions, given that its denominator has a double real root:

$\displaystyle \int \dfrac{5x - 2}{x^2 - 4x + 4} \, dx$

Determine the integral of the following rational function by first decomposing it into a sum of simpler fractions:

$\displaystyle \int \dfrac{x + 7}{x^2 + 4x + 3} \, dx$

In arctangent-type integrals, it is sometimes necessary to manipulate the denominator. Find the following antiderivative by completing the square in the denominator so that it takes the form $\int \frac{u'}{1+u^2} dx$:

$\displaystyle \int \dfrac{3}{x^2 + 2x + 5} \, dx$

It is known that the gradient of the tangent line to a curve at any point $(x, y)$ is given by the expression $m = \dfrac{x^3 + x^2}{x^2 + 1}$. Find the equation of this curve, knowing that it passes through the point $(0, 1)$.

During a spring snowmelt event in the Aragonese Pyrenees, technicians from the Confederación Hidrográfica del Ebro (CHE) monitor the extraordinary water inflow into a reservoir. The net inflow rate is modeled during the first days by the function:

where $t \ge 0$ is the elapsed time in days and $q(t)$ is measured in cubic hectometers per day ($\text{hm}^3/\text{day}$).

1. a) Perform polynomial division to decompose $q(t)$ into the sum of a polynomial and a proper rational fraction with an irreducible denominator. Determine the equation of the slant asymptote of the function and interpret its meaning regarding the long-term trend of the flow rate.
2. b) Find the function $V(t)$ that describes the total extraordinary volume accumulated in the reservoir since the start of the event ($t=0$), knowing that $V(0) = 0$. Clearly show how you decompose the remaining proper fraction into a logarithmic term and another associated with the arctangent.
3. c) The dam's safety protocol establishes that a spillway must be opened preventively if the accumulated volume exceeds $10\text{ hm}^3$ at the end of the second day ($t=2$). Taking the approximations $\ln 5 \approx 1,61$ and $\arctan 2 \approx 1,11\text{ rad}$, numerically evaluate $V(2)$ and justify quantitatively whether the CHE should order the opening of the spillway at that moment.

Consider the family of rational functions depending on a real parameter $m$:

1. a) Obtain the partial fraction decomposition of $f_m(x)$, expressing the coefficients of the numerators in terms of $m$.
2. b) A student puts forward the following conjecture: “By properly adjusting the value of $m$, it is possible to make the antiderivative $\int f_m(x) \, dx$ a purely rational algebraic function, that is, one that contains no logarithmic terms”. Analytically prove that the student's conjecture is FALSE: verify that the sum of the three coefficients of the decomposition is constant and independent of $m$, and deduce why they cannot all vanish simultaneously for any value of $m$.
3. c) Find the only value of $m$ that manages to make the coefficient of the term $\ln|x+2|$ in the antiderivative vanish. For that value of $m$, calculate the antiderivative $F(x)$ that satisfies $F(2) = 0$, write its expression for $x > 1$ by grouping the terms into a single logarithm using its properties, and calculate and interpret the limit $\lim_{x \to 1^+} F(x)$.