Indefinite integrals (direct/u-substitution): notes and practice questions

If the velocity, $v\ ms^{- 1}$, of a particle, starting from the origin, at time, $t$ seconds, is $v(t) = t^{2}cos(t^{3})$.

$a$ Find the displacement equation.

$b$ Find acceleration after 1 second.

A high-speed drone is being tested, and its acceleration, $a$, is modelled by the function $a(t) = \frac{dv}{dt} = -0.8t + 20$, where $v$ is the speed of the drone in m/s and $t$ is the time in seconds, for $0 \le t \le 30$ seconds.

Determine whether the speed of the drone is increasing or decreasing at $t = 28$ seconds.

It is observed that when $t = 5$ seconds, the speed of the drone is $90$ m/s.

Find an expression for the function $v(t)$.

A mathematical model for a physical phenomenon involves the expression $\left(\frac{1}{u} + 2\right)^2$.

(a) Expand $\left(\frac{1}{u} + 2\right)^2$.

The rate of change of a certain quantity is given by $f'(x) = \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4$.

(b) Find the indefinite integral $\int \left( \frac{1}{(x+1)^2} + \frac{4}{x+1} + 4 \right) dx$.

A designer is creating a custom-shaped container. The cross-sectional profile of the container is defined by the curve $y = \frac{1}{x+1} + 2$ for $1 \le x \le 3$. The container is formed by rotating this region $2\pi$ about the x-axis.

(c) Calculate the volume of the solid formed. Give your answer in the form $\frac{\pi}{4}(a + b \ln(c) )$, where $a, b, c \in \mathbb{Z}$.

A chemical reaction is monitored in a laboratory. The concentration of a specific compound, C, in milligrams per litre (mg/L), changes over time, t, in minutes.

The rate of change of the compound's concentration is modelled by

$\frac{\mathrm{d}C}{\mathrm{d}t} = -0.5t + 10, \quad t \ge 0$

At 8 minutes, the concentration of the compound is 70 mg/L.

Find an expression for C in terms of t.

The experiment continues for an extended period.

Describe how the compound's concentration changes if the time elapsed is between 25 minutes and 35 minutes. Justify your answer.

The rate of change of the volume of water in a reservoir, in thousands of cubic meters per year, is given by the equation:

$\frac{\mathrm{d}V}{\mathrm{d}t} = 6t^2 - 20t$

where $V$ is the volume of water in thousands of cubic meters and $t$ is the time in years since the start of monitoring.

(a) Determine whether the volume of water in the reservoir is increasing or decreasing when $t=2$ years.

(b) One year after the start of monitoring, the volume of water in the reservoir was 5 thousand cubic meters. Find an expression for $V(t)$, the volume of water in the reservoir at time $t$, for $t \ge 0$.

Alex is preparing for a calculus exam and encounters a series of integrals. For each integral, identify whether it can be evaluated analytically using standard techniques (and if so, state the appropriate analytical method) or if it requires numerical approximation using technology (e.g., a GDC).

(a) $\int \cos(3x + 2)\,dx$

(b) $\int 2x e^{x^2}\,dx$

(c) $\int x \sin(x)\,dx$

(d) $\int e^{2x^2}\,dx$

(e) $\int_{1}^{2} \frac{4x^2}{x^3 + 3}\,dx$

(f) $\int_{1}^{4} 4\sin(x^2)\,dx$

A colony of bacteria is growing in a nutrient solution. The rate of change of the population, $P$, with respect to time, $t$ (in hours), is modelled by the differential equation $\frac{dP}{dt} = P \cos t (e^{-\sin t})$. At time $t = 0$, the population is $P = 10$.

(a) By using Euler's method with a step length of 0.1, find an approximate value for the population when $t = 0.3$. Give your answer to three significant figures.

(b) By solving the differential equation, find the percentage error in your approximation for the population when $t = 0.3$. Give your answer to three significant figures.

The cross-section of a decorative garden bed is modelled by the curve $y = 2(x-1)(x-3)^2$, where $x$ and $y$ are measured in metres. The garden bed is built on flat ground, represented by the $x$-axis.

(a) Write down the $x$-intercepts of this curve.

(b) Write down a definite integral that represents the area of the cross-section of the garden bed.

(c) Find the value of this area.

(a) A drone's vertical velocity, $v$ metres per second, at time $t$ seconds, is given by $v = t \sin(2t^2)$.

Find an expression for the vertical acceleration of the drone.

(b) Hence, or otherwise, find its greatest vertical acceleration for $0 \le t \le 3$ seconds.

(c) The drone starts at ground level (displacement is 0). Find an expression for the vertical displacement of the drone.

(d) Hence show that the drone never descends below ground level.

A drone is launched vertically upwards from a platform. Its vertical velocity, $v \text{ ms}^{-1}$, at time $t$ seconds, is given by the function:

$v = -3t^2 + 18t - 15$, for $t \ge 0$.

Find the times when the drone is momentarily at rest.

Find the magnitude of the drone's vertical acceleration at $t = 5$ seconds.

Find the greatest speed of the drone in the interval $0 \le t \le 5$.

The drone starts from an initial height of $10$ metres above the ground. Find an expression for the height of the drone, $h$ metres, above the ground at time $t$ seconds.

Find the total distance travelled by the drone in the interval $0 \le t \le 4$.

The rate of profit, $P(t)$, in thousands of dollars per month, for a new product is modelled by the piecewise function

$P(t) = \begin{cases} P_1(t), & 0 \leq t \leq T \\ P_2(t), & t \geq T \end{cases}$

where $P_1(t) = -t^2 + 6t$ and $P_2(t) = -2t + 16$. For a smooth transition in the profit rate, it is required that $P_1(T) = P_2(T)$.

Find the value of $T$.

Show that $P_1'(T) = P_2'(T)$.

The total profit from the product at time $t=0$ is zero.

Find the time when the total profit returns to its initial position.

The rate of change of pollution, $\frac{dP}{dt}$ (in tonnes per day), in a protected lake is modelled by $\frac{dP}{dt} = 1 - 0.1t$, where $t$ is the time in days since monitoring began, for $0 \le t \le 15$.

(i) Find the value of $\frac{dP}{dt}$ at $t = 4$ days.

(ii) Interpret the meaning of your answer to part (a) (i) in context.

Use $\frac{dP}{dt}$ to find the value of $t$ when the pollution in the lake reaches its maximum level.

Two days after monitoring began, the pollution in the lake was $5.5$ tonnes.

Find an expression for $P$ in terms of $t$, for $0 \le t \le 15$.

Hence, find the maximum pollution level in the lake.

A second source of pollution, from a nearby factory, is modelled by $Q = 3(1.15)^t$, where $Q$ is the pollution in tonnes and $t$ is the time in days, for $0 \le t \le 15$.

Write down the initial pollution from this factory.

Each day, the pollution from the factory increases by $p$ %.

Find the value of $p$.

Find the value of $t$ when the pollution from the initial lake source and the factory are the same.

A landscape architect is designing a series of modular planter boxes for an urban garden project. The volume, $V$ cm$^{3}$, of a particular planter box is modelled by the function $V(d) = 30d^2 - d^3$, where $d$ is the depth of the planter in cm.

(a) Use your graphic display calculator to find the value of $d$ that will produce the maximum volume.

(b) Show that the maximum volume of the planter box is $4000$ cm$^{3}$.

The width of the planter is $d$ cm, and the length is $(30 - d)$ cm.

(c) The architect is interested in the total linear dimension $L$, which is defined as the sum of the depth, width, and length of the planter. Hence find the value of $L$ when the volume is maximized.

The 'EcoPack' company designs sustainable packaging. They produce a standard closed rectangular storage container with a length of $10$ cm, a width of $6$ cm, and a height of $4$ cm. The information is shown in the diagram.

Calculate the surface area of the container in cm$^2$.

(b) Calculate the length of the longest internal diagonal of the container.

(c) Each week, EcoPack sells $x$ thousand containers. It is known that $\frac{dP}{dx} = -3x+300$, for $x \ge 0$, where $P$ is the weekly profit, in dollars, from the sale of $x$ thousand containers.

Find the number of containers that should be sold each week to maximize the profit.

(d) The profit from the sale of $30000$ containers is $2500.

Find $P(x)$.

(e) Find the least number of containers which must be sold each week in order to make a profit.

The rate of change of a certain quantity, $Q$, with respect to time, $t$ (in minutes), is modelled by the function $\frac{dQ}{dt} = \frac{10}{5t-2}$.

(a) Find an expression for $Q(t)$ in terms of $t$, assuming an arbitrary constant of integration.

(b) Given that the model is valid for $t > \frac{2}{5}$, find the exact total change in the quantity $Q$ from $t=1$ minute to $t=4$ minutes. Give your answer in the form $a \ln b$, where $a, b \in \mathbb{N}$.

(a) A drone launches a rescue package from an initial position of $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$ metres, relative to an origin on the ground. The package is launched with an initial speed of $15\text{ms}^{-1}$ at an angle $\theta$ to the horizontal ground, where $0 < \theta < \frac{\pi}{2}$.

The velocity components of the package, $t$ seconds after it is launched, are given by $v_x(t) = 15\cos\theta$ and $v_y(t) = 15\sin\theta - 9.8t$.

Find an expression for $x$, the horizontal displacement from the origin, in terms of $\theta$ and $t$.

(b) It is given that the vertical displacement of the package from the ground is $y = (15\sin\theta)t - 4.9t^2$. When the package hits the ground, show that $t = \frac{150\sin\theta}{49}$.

(c) Let $x_g$ be the value of $x$ when the package hits the ground.

Find an expression for $x_g$ in terms of $\theta$ only.

(d) Hence, find the value of $\theta$ which maximizes the value of $x_g$.

(e) The model is adapted to account for a horizontal wind with speed $2\text{ms}^{-1}$ acting in the opposite direction to the initial horizontal motion.

(i) In this new model, the horizontal velocity component is $v_x(t) = 15\cos\theta - 2$. The time taken for the package to hit the ground remains $t = \frac{150\sin\theta}{49}$.

Find an expression for $x_g$, the value of $x$ when the package hits the ground, in terms of $\theta$ only.

(ii) Hence, find the value of $\theta$ which maximizes the value of $x_g$.

The rate of change of the concentration of a certain chemical in a reaction vessel, in mol/L per minute, is given by $\frac{dC}{dt} = \frac{2t}{t^2 + 5}$, where $t$ is the time in minutes.

(a) Use a GDC to find the total change in concentration of the chemical from $t = 0$ minutes to $t = 3$ minutes, giving your answer correct to $3$ significant figures.

(b) Find the general expression for the concentration of the chemical, $C(t)$, if the initial concentration at $t=0$ is not considered. That is, find the indefinite integral $\int \frac{2t}{t^2 + 5}\,dt$.

(c) Hence, using your answer from part (b), find the exact total change in concentration of the chemical from $t = 0$ minutes to $t = 3$ minutes.

A small scientific probe is launched into a dense liquid to collect data. Its descent is affected by gravity, buoyancy, and liquid resistance. The direction downwards is taken to be positive.

Initially, a simple model for the probe's velocity, $v$ ms$^{-1}$, at time $t$ seconds, assumes constant effective acceleration due to gravity and buoyancy, $g'$ ms$^{-2}$, given by:

$\frac{dv}{dt} = g'$

When the probe enters the liquid at $s = 0$ m, its initial velocity is $v = 5$ ms$^{-1}$. The displacement from its initial position is $s$ metres.

(i) Use the chain rule to show that $\frac{dv}{dt} = v\frac{dv}{ds}$.

(ii) Assuming that $g'$ is a constant, solve the differential equation $v\frac{dv}{ds} = g'$ to find $v$ as a function of $s$.

(iii) Using $g' = 9.8$ ms$^{-2}$, determine whether the model predicts that the probe will reach a velocity of $15$ ms$^{-1}$ at some point before it reaches a depth of $s = 200$ m. Justify your answer.

To test the model $\frac{dv}{dt}=g'$, the probe conducted a trial descent, and data for $v$ against $t$ was recorded.

(i) If the model is correct, describe the shape of the graph of $v$ against $t$.

(ii) The observed data showed a graph where the velocity increased rapidly at first, then its rate of increase slowed down, eventually approaching a constant value. Use this observation to comment on the validity of the model in part (a).

An improved model considers liquid resistance, using

$\frac{dv}{dt} = g'-k'v^2$

where $k'$ is a positive constant. You are reminded that initially $s = 0$ and $v = 5$. You may assume that $g' - k'v^2 > 0$.

(i) By using $\frac{dv}{dt} = v\frac{dv}{ds}$, solve the differential equation to find $v$ in terms of $s$, $g'$ and $k'$.

The probe's engineers use the graph of $v$ against $t$ from the trial descent to estimate the value of $k'$.

(ii) The gradient $\frac{dv}{dt}$ is estimated to be $3.05$ ms$^{-2}$ when $v = 15$ ms$^{-1}$. Taking $g'$ to be $9.8$ ms$^{-2}$, use this information to show that the engineers found that $k' = 0.03$.

(iii) Hence, find the value of $v$ predicted by this model, as $s$ tends to infinity.

(iv) Find the upper bound for the velocity according to this model, given that $0 < s \le 200$. Give your answer to four significant figures.

(d) A more refined model for the probe's descent suggests that the rate of change of velocity with respect to displacement is given by $\frac{dv}{ds} = \frac{5000}{(1000-s)^2} - 0.0001 v^2$. Use Euler's method with a step length of $50$ m to estimate the value of $v$ when $s = 200$ m. Take the initial velocity $v = 5$ ms$^{-1}$ at $s = 0$ m.

(i) Suggest one improvement to the use of Euler's method which might increase the accuracy of the prediction of the model.

(ii) Suggest one factor not explicitly considered by the model in part (d) which might lead to a difference between the model's prediction and the data collected.

A researcher is modeling various rates of change in a system. For each given rate of change function, find the original function by determining the indefinite integral.

(a) $\int (3x^2 - 8x + 7)\,dx$

(b) $\int (4\cos(2x) - \sin(3x) )\,dx$

(c) $\int (2e^{5x} + 3e^{-x})\,dx$

(d) $\int (5x - 2)^4\,dx$

(e) $\int \frac{12t^3 - 6t^2}{3t^2}\,dt$