Indefinite integrals + u-substitution: notes and practice questions

By using an appropriate substitution, show that $\int \sin(\sqrt{x}) \, dx = 2\sin(\sqrt{x}) - 2\sqrt{x} \cos(\sqrt{x}) + C$.

The following diagram shows part of the curve $y = \sin(\sqrt{x})$ for $x \ge 0$.

The curve intersects the x-axis at $x_1, x_2, x_3, \dots$.

The nth x-intercept of the curve, $x_n$, is given by $x_n = n^2 \pi^2$, where $n \in \mathbb{Z}^+$.

Write down an expression for $x_{n+1}$.

The regions bounded by the curve and the x-axis are denoted by $R_1, R_2, R_3, \dots$ as shown on the diagram.

Calculate the area of region $R_n$.

Give your answer in the form $(an+b)\pi$, where $a, b \in \mathbb{Z}^+$.

Hence, show that the areas of the regions $R_1, R_2, R_3, \dots$ form an arithmetic sequence.

Expand and simplify $(1+a)^3$ in ascending powers of $a$.

By using a suitable substitution for $a$, show that $1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) = 8\cos^6(x)$.

Consider the function $g(x) = 4\sin(x)(1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) )$.

Show that $\int_0^p g(x) \, dx = \frac{32}{7}(1-\cos^7 p)$, where $p$ is a positive real constant.

It is given that $\int_p^{\frac{\pi}{2}} g(x) \, dx = \frac{1}{28}$, where $0 \le p \le \frac{\pi}{2}$. Find the value of $p$.

Consider the family of integrals defined by $I_n = \int x^n e^{-x} \, dx$ for $n \in \mathbb{N}_0$, where $\mathbb{N}_0 = \{0, 1, 2, ...\}$.

(a) By using integration by parts, show that $I_n = -x^n e^{-x} + n I_{n-1}$ for $n \ge 1$.

(b) Hence, find an explicit expression for $\int x^3 e^{-x} \, dx$.

(c) The region $R$ is enclosed by the graph of $y = x^{3/2} e^{-x/2}$ and the $x$-axis for $x \ge 0$. The region $R$ is rotated by $2\pi$ radians about the $x$-axis. Find the volume of the solid generated.

(d) Show that $\lim_{x \to \infty} x^n e^{-x} = 0$ for any $n \in \mathbb{N}$.

Consider the function $h(x) = x e^{-x}$.

(e) (i) Find the Maclaurin series for $h(x)$ up to and including the term in $x^4$.

(ii) Hence, find the value of the fourth derivative of $h(x)$ at $x=0$, i.e. $h^{(4)}(0)$.

The continuous random variable $X$ has probability density function

$f(x) = \begin{cases} \frac{k}{1+4x^2}, & 0 \le x \le \frac{1}{2} \\ 0, & \text{otherwise.} \end{cases}$

(a) Find the value of $k$.

(b) Find $E(X)$.

Consider the function $f$ defined by $f(x)=\frac{6x^2}{(x^3+1)^2}$ for $x \ge 0$. The graph of $f$ is shown in the following diagram.

Show that $f'(x)=\frac{12x(1-2x^3)}{(x^3+1)^3}$.

Find $\int f(x)dx$.

Consider a function $g(x)$ defined for $x \ge 0$. The derivative of $g$ is such that $g'(x) = f'(x)$, for all $x \ge 0$.

Let $R$ be the region enclosed by the graph of $f$, the graph of $g$, the line $x = 0$ and the line $x = 2$. The area of $R$ is $\frac{16}{3}$.

Find the two possible expressions for $g(x)$.

The rate of change of a certain quantity $R$ with respect to a variable $x$ is given by $R'(x)=\frac{1}{x(M-x)}$, $x \in \mathbb{R}$, $x \neq 0$, $x \neq M$ where $M$ is a positive constant.

The expression for $R'(x)$ can be written in the form $\frac{A}{x} + \frac{B}{M-x}$, where $A, B \in \mathbb{R}$.

Find $A$ and $B$ in terms of $M$.

Hence, find an expression for $R(x)$.

The concentration of a certain chemical product, $C$ (in mol/L), in a reaction vessel at time $t$ (in minutes) can be modelled by the differential equation $\frac{dC}{dt} = \frac{C(L-C)}{8L}$, where $L$ is the maximum possible concentration and $C(0) = 0.2$ mol/L is the initial concentration.

By solving the differential equation, show that $C = \frac{0.2 L}{(L-0.2)e^{-\frac{t}{8}}+0.2}$.

At $t=12$ minutes, the concentration of the product has reached $0.6$ mol/L.

Find the value of $L$, giving your answer correct to four significant figures.

Find the value of $t$ when the rate of change of the concentration is at its maximum.

The growth of a bacterial colony, $B$, in a petri dish can be modelled by the logistic differential equation

$\frac{\text{d}B}{\text{d}t} = k B \left(1 - \frac{B}{N}\right)$

where $t$ is the time measured in hours and $k, N$ are positive constants.

The constant $N$ represents the maximum number of bacteria the petri dish can sustain indefinitely due to limited nutrients.

In the context of this bacterial growth model, interpret the meaning of $\frac{\text{d}B}{\text{d}t}$.

Show that $\frac{\text{d}^2B}{\text{d}t^2} = k^2B\left(1-\frac{B}{N}\right)\left(1-\frac{2B}{N}\right)$.

Hence show that the bacterial colony will grow at its maximum rate when $B = \frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{\text{d}B}{\text{d}t}$ in terms of $k$ and $N$.

Let $B_0$ be the initial number of bacteria.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt = \ln\left(\frac{B(N-B_0)}{B_0(N-B)}\right)$.

After 5 hours, the number of bacteria is $2B_0$. It is known that $N = 3B_0$.

Find the value of $k$ for this bacterial growth model.

A team of engineers is designing a new roller coaster ride. The path of a certain section of the ride can be modelled by the function $g(x)=\frac{x^2 + 3x - 10}{x-4}$, where $x$ is the horizontal distance in metres from the starting point and $g(x)$ is the vertical height in metres. The domain of the function is $x \in \mathbb{R}, x\neq 4$.

Find the coordinates where the path of the roller coaster crosses the horizontal ground (x-axis).

Find the coordinates where the path of the roller coaster crosses the vertical axis (y-axis).

Write down the equation of the vertical asymptote of the graph of $g$.

The oblique asymptote of the graph of $g$ can be written as $y = ax + b$ where $a, b \in \mathbb{Z}$.

Find the value of $a$ and the value of $b$.

Sketch the graph of $g$ for $-20 \le x \le 20$, clearly indicating the points of intersection with each axis and any asymptotes.

Engineers want to analyse the inverse of the roller coaster's height function, $h(x) = \frac{1}{g(x)}$.

Express $h(x)$ in partial fractions.

Hence find the exact value of $\int_{0}^{1} h(x) dx$, expressing your answer as a single logarithm.

(a) In a controlled biological experiment, the rate of change of the population $P$ of a certain microorganism with respect to time $t$ is modeled by the differential equation $t^2 \frac{dP}{dt} = P^2 - 2tP + 2t^2$, where $t > 0$ is in hours and $P$ is in thousands of organisms. It is known that at $t = 1$ hour, the population is $P = 4$ thousand.

Use Euler's method, with a step length of 0.1, to find an approximate value of $P$ when $t = 1.4$.

(b) Use the substitution $P = vt$ to show that $t\frac{dv}{dt} = v^2 - 3v + 2$.

(c.i) By solving the differential equation from part (b), and given that $P > 2t$, show that $P = \frac{6t - 2t^2}{3 - 2t}$.

(c.ii) Find the actual value of $P$ when $t = 1.4$.

(c.iii) Using the graph of $P = \frac{6t - 2t^2}{3 - 2t}$, suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of $P$ at $t = 1.4$.