Semi-immediate integrals
$\displaystyle\int (3x-4)^5\,dx$ is not literally in the table, but it looks very much like it: it is a power, only raised not to $x$ but to $3x-4$. These integrals are called semi-immediate: with a small adjustment of constants, they are solved using the same table as before.
Linear argument: when there is $ax+b$ inside
Calculate the following indefinite integral of a composite power function, making the necessary adjustment to the constant so that the integrand matches the derivative…
Calculate the following indefinite integral of a composite power function, making the necessary adjustment to the constant so that the integrand matches the derivative of the inner function:
$$\int (3x - 4)^5 \, dx$$Find the antiderivative of the exponential function whose derivative requires a small numerical adjustment to the coefficient: $$\int e^{2x + 3} \, dx$$
Find the antiderivative of the exponential function whose derivative requires a small numerical adjustment to the coefficient:
$$\int e^{2x + 3} \, dx$$A content creator has analysed that the growth rate of their followers on a social network follows a marginal function given by $f'(x) = (2x + 5)^3$, where $x$…
A content creator has analysed that the growth rate of their followers on a social network follows a marginal function given by $f'(x) = (2x + 5)^3$, where $x$ represents the time in months. Calculate the family of antiderivatives $F(x)$ describing the total number of followers.
Logarithmic type: when the numerator is (almost) the derivative of the denominator
Determine the indefinite integral of the following rational function. Observe that the numerator is almost the derivative of the denominator, so it is a logarithmic-type…
Determine the indefinite integral of the following rational function. Observe that the numerator is almost the derivative of the denominator, so it is a logarithmic-type integral:
$$\int \frac{1}{5x - 2} \, dx$$In a study of an advertising campaign on social media, it has been determined that the growth rate of an account's followers, $x$ days after the start of the campaign,…
In a study of an advertising campaign on social media, it has been determined that the growth rate of an account's followers, $x$ days after the start of the campaign, is given by the function $f'(x) = \frac{x}{x^2 + 9}$, with $x \geq 0$.
Calculate the general expression of the total followers function $f(x)$, knowing that it is a logarithmic-type integral:
$$\int \frac{x}{x^2 + 9} \, dx$$A courier company determines that the marginal cost of sending a parcel to a distance $x$ (in hundreds of kilometres) is given by the function…
A courier company determines that the marginal cost of sending a parcel to a distance $x$ (in hundreds of kilometres) is given by the function $C'(x) = \frac{2x + 1}{x^2 + x + 4}$. Determine the total cost function $C(x)$, knowing that there are fixed management costs (when $x=0$) of $10$ euros.
Power and exponential type: $u^n\cdot u'$ and $e^u\cdot u'$
The charging rate of a graphene battery in a laboratory behaves according to the function $v(t) = t \cdot e^{t^2}$, where $t$ is time in minutes. Calculate the function…
The charging rate of a graphene battery in a laboratory behaves according to the function $v(t) = t \cdot e^{t^2}$, where $t$ is time in minutes. Calculate the function representing the total accumulated energy $E(t)$, taking into account that at the initial time $t = 0$, the accumulated energy is $5$ units.
Solve the following indefinite integral by identifying the structure of the derivative of a composite function (power type): $$\int (x^2 + 1) \cdot (x^3 + 3x - 2)^4 dx$$
Solve the following indefinite integral by identifying the structure of the derivative of a composite function (power type):
$$\int (x^2 + 1) \cdot (x^3 + 3x - 2)^4 dx$$In a biology laboratory, the spread of a bacterium is studied. The rate of change of the population $P(t)$ as a function of time $t$ (in hours) follows the law: $$P'(t)…
In a biology laboratory, the spread of a bacterium is studied. The rate of change of the population $P(t)$ as a function of time $t$ (in hours) follows the law:
$$P'(t) = \frac{10t^2}{(t^3 + 8)^2}$$If initially ($t=0$) there are $1.000$ bacteria, determine the bacterial population after $2$ hours.
Recognising the pattern before calculating
Before you start calculating, look at the function and identify: which part plays the role of $u(x)$ (what is raised to a power, in the exponent, under the root or in the denominator) and which part plays the role of $u'(x)$ (what goes with it)? If they match up to a constant, it is semi-immediate.
Given the function $f(x) = \frac{e^x}{1 + e^x}$, you are asked to: a) Find its general antiderivative $F(x)$, written in the form $F(x) = \ldots + C$, where $C$ is the…
Given the function $f(x) = \frac{e^x}{1 + e^x}$, you are asked to:
- a) Find its general antiderivative $F(x)$, written in the form $F(x) = \ldots + C$, where $C$ is the constant of integration.
- b) Determine the value of the constant $C$ so that the antiderivative satisfies $F(0) = \ln(2) + 3$.
Calculate the antiderivative of the function $f(x) = \frac{x^2}{\sqrt{1 - x^6}}$ that passes through the origin of coordinates. The figure shows the graph of $f$, which…
Calculate the antiderivative of the function $f(x) = \frac{x^2}{\sqrt{1 - x^6}}$ that passes through the origin of coordinates. The figure shows the graph of $f$, which is defined on $(-1,\,1)$ and has vertical asymptotes at $x=-1$ and $x=1$. Hint: Try to relate the exponent in the numerator to the one in the denominator to identify an arcsine-type structure.