Exercises: Integration by parts

To get started with the integration by parts method, it is essential to know how to choose which function will be $u$ (the one we differentiate) and which will be $dv$ (the one we integrate). Using the mnemonic rule ALPES (Arcs, Logarithms, Polynomials, Exponentials, Sines/Cosines), identify $u$ and $dv$ to solve the following indefinite integral, applying the formula $\int u\,dv = u\cdot v - \int v\,du$:

Find the antiderivative of the function $f(x) = x \cdot \cos(x)$. Remember that when applying the integration by parts method, you should aim for the new resulting integral to be simpler than the original one.

A classic integration by parts example is the logarithmic function, as it does not have a direct standard integral. In this case, we must consider that the function is multiplied by $1$ (that is, $dv = dx$).

Calculate the following indefinite integral:

Find the set of all antiderivatives of the following function, simplifying the final result by factorising if possible:

Calculate the indefinite integral of the function $g(x) = 2x \cdot \sin(x)$. Show step-by-step your choice for $u$ and for $dv$, as well as their corresponding differential of $u$ ($du$) and integral of $v$ ($v$).

The rate of growth of a bacteria population in a laboratory follows the marginal function $B'(t) = \ln(2t)$, where $t$ is the time in hours. Find the general population function $B(t)$ by integrating by parts.

Determine the definite integral over the interval $[1, e]$ of the function $f(x) = 5 \ln(x)$.

Calculate the area of the plane region bounded by the curve $y = x \cdot e^{-x}$, the x-axis $OX$ and the vertical lines $x = 0$ and $x = 1$.

When facing the calculation of antiderivatives, recognizing the structure of the function before operating avoids applying lengthy methods unnecessarily. Look at these two indefinite integrals:

1. a) Justify in detail which of the two integrals is immediate (or solvable using the chain rule) and which necessarily requires the integration by parts method.
2. b) For the integral that requires integration by parts, apply the ALPES rule criterion to justify your choice of $u$ and $dv$, and calculate its family of antiderivatives.

Carla tried to calculate the indefinite integral $\int x \cdot \ln(x) \, dx$ on an exam and wrote down the following working:

"Step 1: I choose $u = x \implies du = dx$ and $dv = \ln(x) \, dx \implies v = \frac{1}{x}$.

Step 2: I apply the formula: $\int x \cdot \ln(x) \, dx = x \cdot \frac{1}{x} - \int \frac{1}{x} \, dx = 1 - \ln|x| + C$".

1. a) Identify and explain, with reasoning, the two conceptual errors made by Carla (one regarding the priority of choice in the LIATE/ALPES rule and the other in obtaining $v$ from $dv$).
2. b) Solve the integral correctly step by step using the integration by parts method, and check your result by differentiating the function obtained.

A well-known influencer has analysed the growth of their followers on a new social network. The instantaneous rate of growth (followers per day) is given by the function $f(t) = (2t + 10) \cdot e^{0,1t}$, where $t$ is time in days since the launch of their channel. Use the method of integration by parts to calculate the function $S(t)$ representing the total number of accumulated followers, knowing that at the time of launch ($t = 0$) the channel had 500 followers.

In a study on learning psychology, it has been determined that the ability to retain new vocabulary terms in a foreign language follows a rate marked by the function $v(t) = t^2 \cdot \ln(t)$, where $t$ is study time in hours ($t \geq 1$). Calculate the antiderivative of this function passing through the point $(1, 2)$.

An audio company is designing a filter to equalise audio tracks. The energy dissipated by a component over a time interval is calculated using the definite integral of the function $E(t) = x \cdot \cos(x)$. Determine the exact value of the area enclosed by the function $f(x) = x \cdot \cos(x)$, the x-axis, and the vertical lines $x = 0$ and $x = \frac{\pi}{2}$.

Calculate the indefinite integral of the function $f(x) = \ln(3x + 2)$ using the integration by parts method. Hint: you can consider that the function is multiplied by 1 to choose your variables $u$ and $dv$.

The marginal profit of a vintage clothing shop (in hundreds of euros per month) is defined by the function $B'(x) = x^2 \cdot e^{-x}$, where $x$ represents the number of months since its opening. Applying the integration by parts method as many times as necessary, find the expression for the total profit $B(x)$, knowing that at the end of the first month ($x = 1$) the profit was zero.

Find the area of the region bounded by the graph of the function $f(x) = (x - 1) \cdot e^x$, the $OX$-axis and the lines $x = 0$ and $x = 1$. Draw a small sketch of the region before calculating the integral.

The marginal cost of production in an e-scooter factory fits the function $C'(x) = \frac{\ln(x)}{x^2}$ for $x > 1$, where $x$ is the number of units produced in thousands. Determine the total cost function $C(x)$ if the fixed costs (cost when $x = 1$) amount to 10 (in thousands of euros).

Calculate the following indefinite integral combining trigonometric and polynomial functions:

Make sure to simplify the final result as much as possible.

We want to calculate the area of the region in the first quadrant bounded by the curve $f(x) = x \cdot \ln(x)$, the x-axis $y = 0$, and the vertical lines $x = 1$ and $x = e$.

1. a) Before integrating, make a reasoned geometric estimation: find an upper bound for the area by calculating the area of the circumscribed rectangle (with base $[1, e]$ and height $f(e)$) and that of the triangle connecting the points $(1, 0)$, $(e, 0)$, and $(e, f(e))$. Justify which of the two bounds is more accurate and calculate its approximate numerical value taking $e \approx 2,72$.
2. b) Determine the exact value of the area by calculating the definite integral $\int_1^e x \cdot \ln(x) \, dx$ using the method of integration by parts and Barrow's rule.
3. c) Express the exact value in decimal form to two decimal places and check whether the result validates and is consistent with your previous geometric estimation.

The National Statistics Institute (INE) tracks the evolution of commercial expenditure during an economic revitalisation campaign. Analysts model the instantaneous rate of change of cumulative expenditure using the function:

where $t$ is the elapsed time in months since the launch ($t \in [0, 6]$) and $g(t)$ is measured in thousands of euros per month.

1. a) Using the integration by parts method, determine the function $G(t)$ that expresses the total cumulative expenditure after $t$ months, knowing that at the start of the campaign ($t = 0$), the cumulative expenditure attributable to it was 0 euros.
2. b) Calculate the total cumulative expenditure upon completing the first 4 months of the campaign ($t = 4$) and interpret what limiting value the expenditure would tend towards if the campaign were maintained indefinitely ($t \to \infty$).

Calculate the antiderivative of the following function passing through the point $(1, 2)$:

Note: Remember that after applying the integration by parts formula, you may need to perform polynomial division or an adjustment to solve the resulting integral.

A mobile application development company has detected that the growth rate of new users for its latest social network follows the function:

where $t$ represents time in months since its launch. Determine the function describing the total number of accumulated users, knowing that at the time of launch ($t=0$) the application had $1000$ beta phase users.

Calculate the value of the following definite integral applying the method of integration by parts:

Find the area of the plane region bounded by the graph of the function $f(x) = (x-1) \cdot e^x$, the x-axis $OX$, and the vertical lines $x = 0$ and $x = 2$. Illustrate the shaded region in your exercise book before carrying out the calculation.

An engineer models the vibration velocity of a structure subjected to forced oscillations using the function (in metres per second):

Calculate the indefinite integral of this function. Note that this type of integral is called "cyclical", since by applying the method of integration by parts twice, the original integral will reappear on the right-hand side of the equality.

Calculate the antiderivative of the function $g(x) = \frac{\ln x}{x^3}$ knowing that its graph has a point of inflection whose $y$-coordinate is equal to $\frac{1}{4}$.

Solve the following integral by using a substitution first and then the method of integration by parts:

Hint: Use the substitution $t = \sqrt{x}$. Do not forget to calculate the new differential $dx$ in terms of $dt$.

Determine the value of the real parameter $k > 0$ knowing that the area of the region enclosed by the curve $f(x) = k \cdot x \cdot e^x$, the $OX$-axis, and the lines $x = 0$ and $x = 1$ is exactly equal to $5$ square units.

The finance department of a tech company estimates that the marginal revenue function resulting from marketing $x$ thousand software licences ($x \ge 1$) follows the model:

where $a$ and $b$ are constant real parameters. The complete formula is unknown, but an external audit provides two pieces of data:

1. The derivative of marginal revenue at $x = 1$ is $I''(1) = 3$.

2. The increase in total revenue when going from 1 to $e$ thousand licences, given by $\int_1^e I'(x) \, dx$, is exactly equal to $\frac{e^2 + 9}{4}$ tens of thousands of euros.

1. a) Reasonably deduce the values of the constants $a$ and $b$. To do this, find the derivative of $I'(x)$ to apply the first piece of data, and solve the definite integral in terms of the parameters using integration by parts to use the second piece of data.
2. b) Once $a$ and $b$ are obtained, calculate the total revenue function $I(x)$ knowing that for the baseline sales level there is no additional cumulative revenue ($I(1) = 0$).

In a study on income distribution in the agri-food sector, the Lorenz Curve, $L(x)$, is modelled, where $x \in [0, 1]$ represents the cumulative proportion of earners and $L(x) \in [0, 1]$ the cumulative proportion of income they receive:

1. a) By definition, every Lorenz Curve must satisfy that $100\,\%$ of earners accumulate $100\,\%$ of the income, that is, $L(1) = 1$. Deduce the value of the parameter $a$ from this condition and check whether the curve passes through the origin $(0, 0)$.
2. b) The Gini Index is given by $I_G = 2 \int_0^1 (x - L(x)) \, dx$. Apply the integration by parts method as many times as necessary to calculate the exact value of this index.
3. c) Knowing that $I_G = 0$ indicates absolute distributive equality and that values above $0,40$ reflect a high concentration of income, approximate your result taking $e \approx 2,718$ and interpret the level of inequality obtained.