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:

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

Aoife 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 Aoife's initial algebraic setup is flawed and what fundamental condition regarding the degrees of polynomials she has ignored.
2. b) Differentiate Aoife'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.

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.

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

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.

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).

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

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

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.

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.

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 $\int \frac{u'}{1+u^2} dx$:

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

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

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

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

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

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