Integration of rational fractions
A marginal cost, an infection rate, a growth speed: many economic and social models are written as a quotient of polynomials $\dfrac{P(x)}{Q(x)}$. That quotient is not in the table of standard integrals, so the job is to turn it into a sum of pieces that are. Which pieces you get depends on the degree of $P$ and on the roots of $Q$.
Prior division: when the numerator is not "smaller"
Before decomposing anything, look at the degrees. If the numerator has degree equal to or greater than the denominator, the fraction is not yet "proper".
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…
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:
$$\int \frac{x + 5}{x + 2} \, dx$$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:…
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.
The numerator is (almost) the derivative of the denominator
Before launching into partial fractions, always check this special case: sometimes the numerator already is (or is very close to) the derivative of the denominator. Then the whole fraction can be integrated in one go, without looking for roots.
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.
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.
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,…
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.
Simple real roots: one fraction for each root
When the numerator is not the derivative of the denominator, factorise the denominator into its roots and set up one fraction for each.
Businesses constantly use the idea of "total cost from marginal cost": if $C'(x)$ is rational, $C(x)$ comes from integrating.
To solve rational integrals using the method of partial fraction decomposition, we must first factorise the denominator. Calculate the following integral knowing that…
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:
$$\int \frac{1}{(x - 2)(x - 5)} dx$$Calculate the indefinite integral of the following rational function by first decomposing it into partial fractions: $$\int \frac{5x - 7}{x^2 - 5x + 6} dx$$ To solve it,…
Calculate the indefinite integral of the following rational function by first decomposing it into partial fractions:
$$\int \frac{5x - 7}{x^2 - 5x + 6} dx$$To solve it, first identify the roots of the denominator and decompose the integrand into the form $\frac{A}{x-x_1} + \frac{B}{x-x_2}$.
Determine the values of the real parameters $A$ and $B$ so that the following partial fraction decomposition equality holds: $$\frac{x + 11}{x^2 - 2x - 15} = \frac{A}{x…
Determine the values of the real parameters $A$ and $B$ so that the following partial fraction decomposition equality holds:
$$\frac{x + 11}{x^2 - 2x - 15} = \frac{A}{x - 5} + \frac{B}{x + 3}$$Once found, calculate the indefinite integral of the original function.
Repeated real roots
If a root is repeated, you need one fraction for each power, not just one.
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 $x = 6$.…
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.
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…
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…
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 $f(x) = \dfrac{x^2 - 5x + 1}{(x+1)(x-2)^2}$.