Find $\displaystyle \int x\sin(cx+d)\;dx,\;\;\int x\cos(cx+d)\;dx $ and hence $\displaystyle \int ax\sin(cx+d)+bx\cos(cx+d)\;dx$

Integrating by parts.

Find $ \int ax\sin(bx+c)\;dx$ or $\int ax e^{bx+c}\;dx$ 

$f(x,y)$ is the PDF of a bivariate distribution $(X,Y)$ on a given rectangular region in $\mathbb{R}^2$.  Write down the limits of the integrations needed to find $P(X \ge a)$, the marginal distributions $f_X(x),\;f_Y(y)$ and the conditional probability $P(Y \le b|X \ge c)$

The random variable $X$ has a PDF which involves a parameter $k$. Find the value of $k$. Find the distribution function $F_X(x)$ and $P(X \lt a)$.

Differentiate $\displaystyle \frac{ax+b}{cx+d}$.

An object moves in a straight line, acceleration given by:

$\displaystyle f(t)=\frac{a}{(1+bt)^n}$. The object starts from rest. Find its maximum speed. 

Solve for $x(t)$, $\displaystyle\frac{dx}{dt}=\frac{a}{(x+b)^n},\;x(0)=0$

Finding the modulus of four complex numbers; includes finding the modulus of a product, a power and a quotient.

$f(X)$ and $g(X)$ as polynomials over the rational numbers $\mathbb{Q}$.

Find their greatest common divisor (GCD) and enter as a normalized polynomial.

A graph shows both the speed and acceleration of a car. Identify which line corresponds to which measurement, and calculate the acceleration during a portion of time.

A simple situational question about a box of chocolates, asking how many of each type there are, what percentage of the box they represent, the probability of picking one and ratios of different types.

Given a geometric sequence, find the common ratio (negative in this question), write down the formula for the nth term and use it to calculate a given term.

Use a piecewise linear graph of speed against time to find the distance travelled by a car.

Finally, use the total distance travelled to find the average speed.

Using the reduction formulas for products of powers or sin and cosine

Practice calculating concentration of solutions

Example of a dilution calculation involving mass concentration and molarity calculations.

Working backwards from the concentration of a diluted solution to find the concentration of "stock" solution.

Practice at a longer calculation to work out how much glucose is needed to make a solution of a given concentration.

Practice calculating number of moles of a substance given the concentration and volume of a solution.

A question to test integration by parts in a "pre-Fourier series" setting.

Newton-Raphson numerical method question to solve $g(x)=0$

Includes a graph of the function $g(x)$ in Advice using Jsxgraph.

Find $\displaystyle \int x(a x ^ 2 + b)^{m}\;dx$

Find $\displaystyle \int \frac{2ax + b}{ax ^ 2 + bx + c}\;dx$

Find $\displaystyle \int \sin(x)(a+ b\cos(x))^{m}\;dx$

Find $\displaystyle \int \frac{a}{(bx+c)^n}\;dx$

Find $\displaystyle \int ae ^ {bx}+ c\sin(dx) + px ^ {q}\;dx$.

Using three -part ratio and amounts of each colour available, calculate how much of a particular shade of paint can be made.

Rotate $y=a(\cos(x)+b)$ by $2\pi$ radians about the $x$-axis between $x=c\pi$ and $x=(c+2)\pi$. Find the volume of revolution.

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Finding areas under graphs using definite integration.

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Find $\displaystyle \int\frac{ax+b}{(1-x^2)^{1/2}} \;dx$. Solution involves inverse trigonometric functions.

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