Exercises: Integration of rational functions

In an investigation into the spread of news on social media, it has been determined that the speed of diffusion follows a rational function. Calculate the antiderivative of the following function, which represents this speed, to find the total impact function:

Calculate the indefinite integral of the function $f(x) = \frac{2x + 4}{x^2 + 4x + 7}$. Hint: Check whether the numerator is the derivative of the denominator.

An electronic components company has calculated that the marginal cost of producing a new model of wireless headphones is given by the function $C'(x) = \frac{3x^2}{x^3 + 10}$, where $x$ is the number of units produced. Determine the total cost function $C(x)$, knowing that fixed costs (when $x=0$) are 500 euros.

To solve rational integrals using the method of partial fraction decomposition, we must first factorise the denominator. Calculate the following integral knowing that the denominator is already factorised:

When the degree of the numerator is greater than or equal to that of the denominator, we must first perform a polynomial division. Perform the division and calculate the integral of the following function:

Find the family of antiderivatives of the rational function $g(x) = \frac{x^2 + 1}{x}$. Hint: Divide each term in the numerator by the denominator.

Calculate the area of the shaded region in the following figure, which is bounded by the function $f(x) = \frac{1}{x}$, the $OX$-axis, and the lines $x = 1$ and $x = e$.

Determine the indefinite integral of the following rational function by adjusting the necessary constants to obtain a natural logarithm:

On an Applied Mathematics for the Social Sciences II exam, students were asked to calculate the following indefinite integral:

Hugo solved the exercise by immediately writing:

justifying on his paper that «whenever there is a polynomial in the denominator, the integral is its natural logarithm».

1. a) Explain in detail Hugo's conceptual error. State what exact mathematical relationship must exist between the numerator and the denominator for a natural logarithm to appear directly, and what is missing in the numerator of this integral to satisfy it.
2. b) Show how the integral should be solved correctly by decomposing the fraction into a sum of partial fractions with simple real roots, determining the coefficients step by step and calculating the final antiderivative.

Consider the following two rational integrals:

A classmate claims that, to solve either of the two, it is mandatory to calculate the roots of the denominator ($x = 2$ and $x = 3$) and set up the partial fraction decomposition $\frac{A}{x-2} + \frac{B}{x-3}$.

1. a) Analyze the structure of both integrands and explain why her claim is unnecessarily long for $I_1$. What direct method allows you to solve $I_1$ in a single step by adjusting a constant if necessary?
2. b) Justify why for $I_2$ it is indeed essential to resort to partial fraction decomposition.
3. c) Solve both integrals with full reasoning, applying the most efficient method in each case.

Calculate the indefinite integral of the following rational function by first decomposing it into partial fractions:

To solve it, first identify the roots of the denominator and decompose the integrand into the form $\frac{A}{x-x_1} + \frac{B}{x-x_2}$.

Find the antiderivative $F(x)$ of the function $f(x) = \frac{x + 4}{x^2 + 4x}$ that satisfies the condition of passing through the point $(1, \ln(5))$.

Remember that to determine the constant of integration $C$, you must use the value of the function at the given point after performing integration by partial fractions.

In a digital marketing campaign study, it has been determined that the rate of acquiring new users follows the function $v(t) = \frac{t^2 + 2t + 3}{t + 1}$, where $t$ is the time in days.

Calculate the total users function $U(t)$, knowing that at the initial moment ($t=0$) there were 100 users.

Calculate the value of the following definite integral using Barrow's rule:

To do so, decompose the fraction into a sum of partial fractions and use the properties of logarithms to simplify the final result.

Calculate the area of the region bounded by the graph of the function $f(x) = \frac{x - 3}{x^2 - 6x + 9}$, the x-axis $OX$, and the vertical lines $x = 4$ and $x = 6$.

Note that the denominator has a double real root. Use the appropriate partial fraction decomposition for this case.

The marginal cost of production for an electronic components company is given by the function $C'(x) = \frac{400}{x + 50}$ (in euros per unit), where $x$ represents the number of units manufactured.

Determine the total cost increase if production increases from 150 to 250 units.

Determine the values of the real parameters $A$ and $B$ so that the following partial fraction decomposition equality holds:

Once found, calculate the indefinite integral of the original function.

Find the area of the region of the plane enclosed between the curve $y = \frac{x^2}{x-1}$ and the line $y = \frac{1}{2}x + 3$.

Follow these steps:

1. Find the intersection points between the curve and the line.

2. Set up the definite integral of the difference between both functions between those points.

3. Solve the resulting rational integral (you will need to perform polynomial division).

An e-commerce consulting firm models the daily conversion rate of an advertising campaign during its first three days using the rational function $f(x) = \frac{x + 5}{x + 2}$, where $x \in [0, 3]$ represents time in days. We wish to calculate the accumulated conversion given by the area under the curve $\int_0^3 f(x)\,dx$.

1. a) Before integrating, make a bounded estimate of the result: justify with reasoning whether $f(x)$ is increasing or decreasing on $[0, 3]$, calculate its extreme values $f(0)$ and $f(3)$, and use these values to obtain a lower bound and an upper bound for the value of the integral (approximating the area using inscribed and circumscribed rectangles).
2. b) Given that $\text{degree}(P) \geq \text{degree}(Q)$, perform polynomial division to express $\frac{x + 5}{x + 2}$ in the form $C(x) + \frac{R(x)}{Q(x)}$.
3. c) Find the antiderivative $F(x)$ and calculate the exact value of the definite integral using Barrow's rule. Check whether the value obtained is consistent with your previous estimate, given that $\ln(5/2) \approx 0,916$.

The Regional Transport Consortium studies the net passenger inflow to an intermodal station during the morning rush hour. The net inflow rate (in thousands of passengers per hour) is given by the function:

where $t \in [0, 3]$ measures the time in hours elapsed since 06:30 in the morning.

1. a) Identify the roots of the denominator and decompose the rate $f(t)$ into a sum of partial fractions, explaining in detail how you obtain the constants.
2. b) Determine the function $N(t)$ that expresses the accumulated number of passengers (in thousands) who have entered the station after $t$ hours, knowing that at 06:30 ($t = 0$) the cumulative counter starts at zero, that is, $N(0) = 0$.
3. c) Calculate how many thousands of passengers will have entered in total by 08:30 ($t = 2$), simplifying the resulting expression using the properties of logarithms and rounding to two decimal places using the data $\ln 2 \approx 0,693$ and $\ln 3 \approx 1,099$.

A courier company has determined that the marginal cost of processing $x$ thousand packages daily is given by the function $C'(x) = \dfrac{x^3 + 2x^2 + 5}{x^2 + 2x}$ (in hundreds of euros). Knowing that the cost of processing 1000 packages daily ($x = 1$) is 1200 euros, find the total cost function $C(x)$.

Calculate the indefinite integral of the function $f(x) = \dfrac{2x^2 - 10x + 7}{(x-1)^2(x-3)}$. Note that the denominator has a double real root and a simple one, which requires a decomposition into three partial fractions.

It is estimated that the rate of change of a city's population, with respect to time $t$ (in years), follows the model $P'(t) = \dfrac{4t^2}{t^2 - 5t + 6}$. If the population in year $t=4$ is 50 000 inhabitants, calculate the expression for the population function $P(t)$ for $t>3$ and determine the total growth experienced between year $t=4$ and year $t=5$.

Find an antiderivative $F(x)$ of $f(x) = \dfrac{x^4 - 3x^2 + 2x - 1}{x^2 - 4}$ satisfying the condition that its graph passes through the point $(3, \ln 5)$. Hint: Perform polynomial division first.

Calculate the value of the definite integral $\displaystyle \int_{0}^{1} \dfrac{e^{2x}}{e^x + 2} \, dx$ using the substitution $e^x = t$. Once the substitution is made, solve the resulting rational integral.

Given the function $f(x) = \dfrac{1}{x^3 - x}$, determine the area of the region bounded by the curve, the x-axis, and the vertical lines $x = 2$ and $x = 3$. Express the final result using a single logarithm.

Determine the values of the parameters $A$, $B$, and $C$ so that the function $F(x) = A \cdot \ln|x+1| + B \cdot \ln|x-2| + \dfrac{C}{x-2}$ is an antiderivative of $f(x) = \dfrac{x^2 - 5x + 1}{(x+1)(x-2)^2}$.

The marginal revenue from selling a new smartphone model (in hundreds of euros per unit) is given by $I'(x) = \dfrac{200x + 400}{x^2 + 4x + 4}$. If the total revenue from selling 8 units is 1 500 euros, what will the total revenue be if 18 units are sold? Note: Observe whether the denominator can be simplified or expressed as a power of a binomial.

Consider a rational function of the form $f(x) = \frac{a x^2 + b x + 5}{(x - 1)^2}$, defined for all $x \neq 1$, where $a$ and $b$ are real coefficients. It is known that an antiderivative of it, $F(x)$, simultaneously satisfies three conditions:

1. Its asymptotic behavior at infinity satisfies $\lim_{x \to +\infty} [F(x) - (2x + 6)] = 0$.

2. The function $f(x)$ admits a partial fraction decomposition of the form $f(x) = a + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}$.

3. The graph of $F(x)$ passes through the point $(2, 7)$.

1. a) Deduce with reasoning the value of the coefficient $a$ and prove that necessarily $B = 0$ by analyzing the general form of the integral of $f(x)$ and the asymptotic behavior of the logarithmic terms as $x \to +\infty$.
2. b) From the values found, algebraically determine the values of the parameters $b$ and $C$ so that the equality between the expressions of $f(x)$ holds for all $x \neq 1$.
3. c) Find the explicit analytical expression of the antiderivative $F(x)$ that satisfies all the given conditions.

Consider the family of indefinite integrals $I(k) = \int \frac{x + k}{x^2 - 4x + 4}\,dx$, for the interval $x > 2$, where $k$ is a constant real parameter.

1. a) Without performing the full partial fraction decomposition, explain for what numerical value of the parameter $k$ the integrand becomes a purely logarithmic-type function (that is, where the numerator is proportional to the derivative of the denominator) and does not generate any power term after integrating. Calculate the antiderivative for that particular value of $k$.
2. b) For any other value $k \neq -2$, compare with reasoning two possible algebraic methods to solve the integral:

• Method A: Formal decomposition by undetermined coefficients: $\frac{x+k}{(x-2)^2} = \frac{A}{x-2} + \frac{B}{(x-2)^2}$.

• Method B: Rewrite the numerator as $(x - 2) + (k + 2)$ and directly separate into two fractions.

State which of the two methods is more agile and justify why both lead to exactly the same result.

1. c) Obtain the general family of antiderivatives $F(x)$ in terms of $k$. Then, analyze with reasoning what sign the limit $\lim_{x \to 2^+} F(x)$ takes depending on whether $k > -2$, $k = -2$, or $k < -2$, justifying which term dominates the behavior at the vertical asymptote.