Integration of rational fractions
A quotient of polynomials $\dfrac{P(x)}{Q(x)}$ is not in the table of standard integrals. The trick is always the same: turn it into a sum of pieces that are in the table. Which pieces appear depends on the degree of $P$ and on the roots of $Q$.
If the degree of the numerator is not smaller: divide first
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…
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.
Simple real roots: one fraction for each root
Calculate the following indefinite integral, observing that the numerator is, or can easily be transformed into, the derivative of the denominator: $$\int \frac{2x +…
Calculate the following indefinite integral, observing that the numerator is, or can easily be transformed into, the derivative of the denominator:
$$\int \frac{2x + 3}{x^2 + 3x + 10} \, dx$$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…
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).
Repeated real roots: one fraction for each power
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…
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 integral of the following rational function by first decomposing it into a sum of simpler fractions: $\displaystyle \int \dfrac{x + 7}{x^2 + 4x + 3} \, dx$
Determine the integral of the following rational function by first decomposing it into a sum of simpler fractions:
$\displaystyle \int \dfrac{x + 7}{x^2 + 4x + 3} \, dx$
Complex roots: an arctangent appears
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…
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…
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.