Exercises: Indefinite integral. Elementary decomposition and standard integrals

A professional skater is analyzing the trajectory of his latest jump on a ramp. The function that describes the variation of his height with respect to time (vertical velocity) is given by the expression $v(t) = 3t^2 - 4t + 5$, whose graph appears in the figure. Find the family of functions $H(t)$ that represent the skater's height, by calculating the indefinite integral:

In a study on how news spreads on a social network, it was determined that the growth rate of "shares" follows a power law. To find the cumulative total, a programmer must solve the following integral of an irrational function. Calculate it by first expressing the root as a fractional exponent:

Determine whether the function $F(x) = x \times e^x$ is an antiderivative of the function $f(x) = (x+1)e^x$. To justify your answer, remember to use the fundamental relationship between a function and its antiderivative via differentiation.

A hearing study for teenagers analyses the intensity of a sound signal. The function describing the instantaneous sound pressure is $p(x) = \frac{3}{x} + e^{2x}$. Find the general antiderivative of this function by carrying out the integration:

Calculate the indefinite integral of the following combination of trigonometric functions and constants, applying linearity properties (factoring out constants and splitting sums):

Find all antiderivatives of the function $f(x) = \frac{x^4 - 2x^2 + 5}{x^2}$.

In the design of a smartphone camera lens, the curvature function needs to be known. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the expression for the function $g(x)$ by evaluating its indefinite integral:

Find the antiderivative of the function $f(x) = 2x - 6$ whose graph passes through the point $P(3, 4)$.

Note: First calculate the indefinite integral to obtain the family of functions with the constant $C$, and then use the point $P$ to find the specific value of that constant.

Cian has to calculate the indefinite integral $\int (3x - 2)(2x + 1)\, dx$ in an exam. On his answer sheet, he writes the following:

1. a) Explain in detail the conceptual error Cian makes when applying the linearity property of the integral.
2. b) Prove that Cian's result is incorrect by differentiating the function he obtained and checking that it does not match the integrand $(3x - 2)(2x + 1)$.
3. c) Solve the integral correctly by first carrying out the algebraic multiplication and then applying the rules for standard integrals.

Consider the continuous function $f(x) = 6 - 2x$ and the specific antiderivative $F(x)$ that satisfies the initial condition $F(0) = 2$.

1. a) Estimate: Without calculating the analytical expression of $F(x)$, analyze with reasoning the sign of $f(x)$ on the interval $[0, 3]$ and on the interval $[3, 4]$. Based on the increasing and decreasing behavior of $F(x)$, and taking into account that $f(x)$ is symmetric with respect to its root $x = 3$, estimate whether the value of $F(4)$ will be greater than, less than, or equal to $F(0)$.
2. b) Calculate: Obtain the general family of antiderivatives of $f(x)$ by elementary decomposition and identify the particular antiderivative $F(x)$ that passes through $(0, 2)$.
3. c) Check: Calculate the exact value of $F(4)$ by substituting into the function found and confirm whether your qualitative estimate was correct.

Find the family of antiderivatives of the following function, simplifying the result by eliminating negative exponents and using radicals:

$f(x) = \dfrac{3}{x^4} - \dfrac{2}{\sqrt[3]{x^2}} + \dfrac{1}{5x}$

Find the indefinite integral of the function $g(x)$, making the necessary constant adjustments to obtain a logarithmic form:

$I = \displaystyle\int \dfrac{3x - 1}{x^2 - \frac{2}{3}x + 5} \, dx$

A surveillance drone moves such that its horizontal velocity (in m/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial instant $t=0$ the drone is at position $s=0$, determine the expression for the position function $s(t)$.

Find the antiderivative of the function $f(x) = \dfrac{x^2 - 4x + 1}{x-2}$ whose graph passes through the point $P(3, 5)$.

Note: Perform polynomial division first.

Determine the indefinite integral of the following composite function, identifying the derivative of the inner function:

$I = \displaystyle\int \dfrac{\ln^2(x) + 1}{x} \, dx$

Find the antiderivative $F(x)$ of the function $f(x) = \dfrac{1}{\cos^2(3x)} + 5^{2x}$ knowing that $F(0) = 1$.

Calculate the indefinite integral of the following function using linearity decomposition and adjusting for the derivative of the exponent:

$I = \displaystyle\int \left( 2x \times e^{x^2+3} - \dfrac{5}{1+4x^2} \right) \, dx$

Given the function $f(x) = \dfrac{\sin(x)}{\sqrt{1 + \cos(x)}}$, calculate its general antiderivative. Check your result by differentiating the obtained function.

Determine the real coefficients $a$ and $b$ of the function $f(x) = a\times x + \dfrac{b}{x^2}$ (defined for $x > 0$) knowing that its family of antiderivatives satisfies the following two conditions:

1. The tangent line to the graph of any of its antiderivatives at the point with abscissa $x = 1$ is horizontal (zero slope).

2. The specific antiderivative $F_0(x)$ that passes through the point $(1, 7)$ also satisfies $F_0(2) = 11$.

1. a) Set up the system of equations that allows you to find the values of $a$ and $b$, interpreting geometrically the relationship between the derivative of an antiderivative and the integrand function.
2. b) Find the function $f(x)$ and the complete analytical expression of the particular antiderivative $F_0(x)$.

In a wind farm located in County Galway, operation technicians model the net power generated, that is, the instantaneous rate of change of the accumulated energy, $P'(t)$ (in $\text{MW}$), during a 9-hour monitoring interval ($t \in [1, 9]$) using the function:

where $t$ is the elapsed time in hours.

1. a) Algebraically decompose the quotient into a sum of elementary powers of the form $k\times t^n$ and state which standard integration rule applies to each term.
2. b) Find the complete family of antiderivatives $P(t)$ that describe the accumulated energy as a function of time.
3. c) If at the first hour of measurement ($t = 1$) the meters recorded an accumulated energy of $15\text{ MWh}$, find the integration constant $C$ and determine the accumulated energy when $4$ hours have elapsed ($t = 4$).

A sound engineer is analysing the signal of a new synthesiser. The rate of change of the wave intensity with respect to time, $t$, is given by the function $f'(t) = \dfrac{e^{2t} + 3e^t}{e^t + 1}$.

Knowing that for a time $t = 0$ the intensity of the wave is $10$ units, determine the algebraic expression for the intensity $f(t)$ by performing the necessary algebraic manipulations to solve the integral.

Find the family of antiderivatives of the following trigonometric function using fundamental identities to transform the integrand into a sum of standard integrals:

Determine the function $f(x)$ whose second derivative is $f''(x) = \dfrac{12x^2 - 4}{(x^2 + 1)^3}$.

It is known that the graph of $f(x)$ has an inflection point with positive $x$-coordinate at which the tangent line is horizontal, and that the graph of $f(x)$ passes through the point $(0, 5)$.

Note: This exercise requires an expert handling of the chain rule in reverse for rational functions.

A logistics company studies the tire wear of its fleet. The wear rate $V(x)$ (in mm per 1000 km) depends on the load $x$ (in tonnes) according to the function:

$V(x) = \frac{x + 1}{\sqrt{4 - (x - 3)^2}}$

Find the accumulated wear function $D(x)$, knowing that for a load of $3$ tonnes ($x = 3$) the wear is zero.

Find the indefinite integral of the following rational function by completing the square in the denominator to identify an arctangent form:

Given the function $f(x) = |x - 2| \times e^x$, find the antiderivative $F(x)$ satisfying $F(0) = 1$.

Remember that for $F(x)$ to be an antiderivative on its entire domain, it must be a continuous function, so you will need to adjust the constants of integration across the different intervals.

Determine the values of the parameters $A, B$, and $C$ such that the function $F(x) = (Ax^2 + B) \sin(2x) + Cx \cos(2x)$ is an antiderivative of $f(x) = x^2 \cos(2x)$.

Set up the system of equations by differentiating $F(x)$ and equating coefficients.

Calculate the indefinite integral of the function:

This exercise assesses the ability to recognize the structure $g(u(x)) \times u'(x)$ when there are multiple composite functions.

Two 6th Year Leaving Certificate students, Aoife and Hugo, debate the most appropriate method to solve the indefinite integral $I = \int \frac{x^2 + 3}{x^2 + 1}\, dx$:

• Aoife argues that polynomial Euclidean division must always be performed to express the improper fraction in the form $\frac{P(x)}{Q(x)} = C(x) + \frac{R(x)}{Q(x)}$.

• Hugo claims that it is faster to use the algebraic trick of adding and subtracting $1$ in the numerator: $x^2 + 3 = (x^2 + 1) + 2$, to directly split the fraction into two standard integrals.

• A third classmate, Daniel, proposes moving the denominator up as $(x^2 + 1)^{-1}$ and directly applying the power rule: $\int u^n = \frac{u^{n+1}}{n+1}$.

1. a) Rigorously justify why Daniel's proposal is a serious mathematical error, indicating what differential element the integrand is missing in order to apply a power-type or logarithmic rule.
2. b) Fully carry out Aoife's and Hugo's procedures, showing that both lead to the same decomposition. Reason which of the two methods is more efficient and carries less risk of error in this particular case.
3. c) Calculate the particular antiderivative $F(x)$ that satisfies the condition $F(1) = \frac{\pi}{2} + 3$.

We wish to analytically reconstruct a function $f(x)$ and one of its antiderivatives, $F(x)$, from the following geometric properties observed in the family of curves:

1. The tangent line to any antiderivative at a point with abscissa $x$ has strictly positive slope if $x \in (-\infty, 0) \cup (2, +\infty)$ and strictly negative slope if $x \in (0, 2)$.

2. The second derivative of any antiderivative is a linear function $F''(x) = \alpha x + \beta$ whose graph intersects the x-axis at $x = 1$.

3. There is a specific antiderivative $F_1(x)$ whose relative minimum is reached at a height of $-3$ units and which also intersects the X-axis at the point $(3, 0)$.

1. a) Translate conditions 1 and 2 with reasoning to prove that the integrand function must be a quadratic polynomial of the form $f(x) = k(x^2 - 2x)$ with $k > 0$.
2. b) Obtain the family of antiderivatives $F(x)$ via elementary decomposition in terms of the parameter $k$ and the constant $C$.
3. c) Use condition 3 to uniquely deduce the values of $k$ and $C$, and write the final formulas for $f(x)$ and for the particular antiderivative $F_1(x)$.