Integration by parts (+ repeated by parts): notes and practice questions

Find $\int x^2 \sin(2x) dx$.

Hence, find the exact value of $\int_0^{\frac{\pi}{4}} x^2 \sin(2x) dx$.

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.

Find $\int x^2 \cos(2x) dx$.

Hence, find the exact value of $\int_0^{\frac{\pi}{4}} x^2 \cos(2x) dx$.

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. $h^{(4)}(0)$.

Find $\int x \sec^2 x \, dx$.

The region $R$ is enclosed by the curve $y = x \sec^2 x$, the $x$-axis, and the lines $x=0$ and $x=\frac{\pi}{3}$. Show that the area of $R$ is $\frac{\pi\sqrt{3}}{3} - \ln 2$.

A pharmaceutical company is testing a new drug. The concentration of the drug, $C$, in the bloodstream of a patient, in micrograms per millilitre ($\mu\text{g/mL}$), $t$ hours after administration, is modelled by the function $C(t) = 5te^{-0.2t}$, for $0 \le t \le 15$.

Sketch the graph of $C(t)$ for $0 \le t \le 15$, clearly indicating the coordinates of the initial concentration point $A$, the maximum concentration point $M$, and the concentration point $B$ at $t=15$ hours.

State the range of the concentration $C(t)$ during the observed period.

Find the equation of the straight line connecting the initial concentration point $A$ and the concentration point $B$ at $t=15$ hours.

Show that the rate of change of the drug concentration is given by $C'(t) = (5-t)e^{-0.2t}$.

At a certain time, the rate of change of the drug concentration is parallel to the line AB. Find the equation of the tangent line to the graph of $C(t)$ at this time. Give all coefficients in your equation correct to $3$ significant figures.

Calculate the area of the region enclosed by the graph of $C(t)$ and the line AB.

Find the general solution to the following differential equation. (a)

$\frac{dy}{dx} = \frac{y^2}{x^2-1}$

(b)

$\frac{dy}{dx} = y^2 \sin x \cos x$

(c) Find the particular solution to the differential equation $(x^2+4) \frac{dy}{dx} = xy$, given the initial condition $y(0)=1$.

(d)

$e^x \frac{dy}{dx} = \frac{1}{y}$

(e)

$\frac{dy}{dx} + 2y = xe^{-x}$

(f) $x \frac{dy}{dx} - 3y = x^5$ for $x>0$.

(g) $\frac{dy}{dx} + y \cot x = \cos x$ for $0 < x < \pi$.

The lifetime, $T$ (in years), of a certain electronic component is modelled by a continuous random variable with probability density function $f(t)$ given by

$f(t) = \begin{cases} kt^2 e^{-t/5} & t \ge 0 \\ 0 & \text{otherwise} \end{cases}$

where $k$ is a positive constant.

(a) Show that $k = \frac{1}{250}$.

(b) Find the mode of the distribution of $T$.

(c) Find the mean lifetime of the component, $E(T)$.

(d) Find the variance of $T$.

A continuous random variable $X$ has probability density function

$f(x) = \begin{cases} kx e^{-x} & x \ge 0 \\ 0 & \text{otherwise} \end{cases}$

Show that $k = 1$.

Find the variance of $X$.