Further Integration (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

Find the integral

$\int \sin x \mathrm{d}x$

Use calculus to evaluate

${\int }_1^4\frac{1}{x} \mathrm{d}x$

Find an expression for $y$ given that

$y=\int 7e^{7x} \mathrm{d}x$

Integrate

$\int \cos 2x \mathrm{d}x$

Use calculus to find the value of

${\int }_0^2(3x-1{)}^3 \mathrm{d}x$

Find an expression for $y$ given that

$\frac{\mathrm{d}y}{\mathrm{d}x}=e^{5x}$

Integrate

$\int \frac{1}{2x+3} \mathrm{d}x$

Find an expression for $y$given that

$\frac{\mathrm{d}y}{\mathrm{d}x}=3e^{3x+1}$

Find an expression for $\mathrm{f}\left(x\right)$ given that

$\mathrm{f}\left(x\right)=\int \sin (2x+1) \mathrm{d}x$

Find an expression for $y$ given that

$\frac{\mathrm{d}y}{\mathrm{d}x}=3\cos (3x+5)$

Use calculus to find the exact value of

${\int }_{\frac{7\pi }{6}}^{\frac{4\pi }{3}}{\sec }^2(x-\pi ) \mathrm{d}x$

Given that

$\cos 2\theta \equiv 2{\cos }^2\theta -1$

Use calculus to find the exact value of

${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\cos }^2\theta d\theta$

Find

$\int \sqrt{1+{\cot }^{2}x} dx$

Use the identity

$\sin 2A\equiv 2\sin A\cos A$

to show that

$4\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)\equiv 2\sin \theta$

Hence, or otherwise, find the integral

$\int 4\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right) \mathrm{d}\theta$

Show that

${\int }_2^{5}4e^{2x-4} \mathrm{d}x=2(e^6-1)$

The diagram below shows part of the graph with equation $y=2^{\ln x}.$

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

${\int }_5^{10}{2}^{\ln x} \mathrm{d}x$

Given that 4 strips are to be used, calculate $h$, the width of each strip.

Complete the table of values below, giving each entry correct to three significant figures.

Find an estimate of the shaded area using the values from the table in part (b).

State whether your answer to part (c) is an overestimate or an underestimate.

The diagram below shows the graph with equation $y=\sqrt{e^{-x}}+2x.$

The area shaded is to be estimated using the trapezium rule where $h=1$.

(i) Write down the number of strips to be used.
(ii) Write down the number of ordinates to be used.

Apply the trapezium rule as described above to estimate the shaded area, giving your answer to three significant figures.

Describe a way in which the estimate calculated in part (b) could be improved.

The energy company Power $x$ operate a wind turbine.
Engineers from Power $x$ model the power output, $P$ kW (kiloWatts), of the wind turbine over a twelve-hour period according to the function

$P\left(t\right)=50+10\sin \left(\sqrt{10t}\right),0\le t\le 12$

where t is time measured in hours. The graph of $y=P\left(t\right)$ is shown below.

Using the trapezium rule, with six strips of width 1, estimate the total power generated by the wind turbine in the first six hours.
You may use the table below to help.

Briefly explain why it is difficult for the engineers from Power $x$to determine whether the total amount of power found, using the method in part (a), is an over- or under-estimate.

Following an explosion on the Longwater drilling rig, oil began to leak into the ocean from a damaged underwater pipe.
It took several months for experts to seal the pipe and stop further oil from leaking.

For the first fifteen weeks of the oil spill, the rate of oil leaking from the pipe (measured in barrels per week) was recorded every three weeks.
The results are shown in the table below.

The trapezium rule is to be used to estimate the total amount of oil leaked during the first fifteen weeks of the spillage. All the data in the table is to be used.
(i) Write down the number of strips ( $n$) that will be used.
(ii) Write down the width of each strip ( $h$) that will be used.

Show that the trapezium rule described in part (a) estimates the amount of oil spilled in the first fifteen weeks is 208 000 barrels to three significant figures.

The leakage rate of the oil rose steadily for the first 15 weeks.
State whether the estimate in part (b) is an under- or an over-estimate.

Coronation Street has a speed limit of 40 mph and traffic is monitored along one stretch of the road by four average speed cameras.

Vera is driving her car along Coronation Street and passes the first camera at time zero. Vera’s speed, measured in miles per hour, and the time she passes each camera, are recorded.

The results are shown in the table below:

Use the trapezium rule with all the data in the table to estimate the distance between the first and last camera.
Give your estimate to three significant figures.

Vera’s car consumes fuel at a rate of 52.6 miles per gallon.
Estimate the amount of fuel, in gallons, Vera’s car uses between the first and last camera.

Use calculus to find

$\int \left({\mathrm{x}}^2+5-\frac{3}{{\mathrm{x}}^2}\right)\mathrm{dx}$

Write down

$\int 3e^{3x}dx$

Find the integral

$\int 5\left({\mathrm{e}}^{5\mathrm{x}}-{\mathrm{e}}^{-5\mathrm{x}}\right)\mathrm{dx}$

Find an expression for y given that

y =$\int \left(\sin x+\cos x\right)dx$

Use calculus to evaluate

${\int }_{-8}^{-2}\frac{1}{x}dx$

giving your answer as a single logarithm.

Integrate

$\int 2\mathrm{sinx}\mathrm{cosx}\mathrm{dx}$

Use calculus to find the value of

${\int }_1^3{\left(4x+1\right)}^{5}dx$

Find an expression for f(x) given that

f(x) = $\int \frac{5}{2x+3}\mathrm{dx}$

Find an expression for y given that

$\frac{dy}{dx}=2e^{3x-5}-5e^{5x}$

Show that

${\sin }^2\left(3x+2\right)\equiv \frac{1}{2}\left(1-\cos \left(6x+4\right)\right)$

Hence, or otherwise, find an expression for

$\int {\sin }^2\left(3x+2\right)dx$

Show that, for $\theta \ne \mathrm{k\pi }$ (where k is an integer),

$\frac{2-2{\cos }^2\theta }{{\sin }^{\mathit{2}}\theta }=\tan \theta$

Use calculus and your result from part (a) to show that

${\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}\frac{2-2{\cos }^2\theta }{\sin 2\theta }\mathrm{d}\theta =\frac{1}{2}\mathrm{In}3$

Use calculus to find the exact value of

${\int }_{\frac{\mathrm{\pi }}{2}}^{\frac{5\mathrm{\pi }}{6}}\frac{2\cos x}{1-\cos 2x}\mathrm{dx}$

Write e^2x(1+e^2x+e²) in the form e^f(x)+e^g(x)+e^h(x), where f(x), g(x), and h(x) are all linear functions of x.

Hence show that

${\int }_0^1{\mathrm{e}}^{2\mathrm{x}}\left(1+{\mathrm{e}}^{2\mathrm{x}}+{\mathrm{e}}^2\right)\mathrm{dx}=\frac{3}{4}\left({\mathrm{e}}^4-1\right)$

The diagram below shows part of the graph with equation y = $3x-e^{x^2}$

Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

Use two separate diagrams to show how the trapezium rule can lead to an underestimate or an overestimate when used to estimate the area under a curve.

Use the trapezium rule with h = 0.25 to find an estimate for the area bounded by the curve with equation y = 1 + 0.3x² sin x, the lines with equations x = 1 and x = 2 and the x-axis.

Give your answer to three significant figures.

The integral

${\int }_1^2\left(1+0.3{\mathrm{x}}^2\mathrm{sinx}\right)\mathrm{dx}$

can be evaluated exactly by applying the method of integration by parts (twice).
Suggest a reason why it may be preferrable to use a numerical method, such as trapezium rule, to estimate the integral rather than use integration by parts to find its exact value.

Use the two diagrams below to show how rectangles can be used to give an upper and lower bound when estimating the area under a curve using the trapezium rule.

The energy company Power^Y own and maintain a wind farm.

Engineers from Power^Y model the power output, P kW (kiloWatts), of a single wind turbine over a twelve-hour period according to the function

P(t) = 50 + 30e^-0.1t sin t 0≤ t ≤ 12

where t is time measured in hours. The graph of y = P(t) is shown below.

Use the trapezium rule with 5 ordinates to estimate the total power generated by the wind turbine in the last four hours of the twelve-hour period.

The drilling rig AlphaBeta began leaking oil into the North Sea following a technical fault. It took 14 hours for engineers to trace and repair the fault.

During that time the rate of oil leaking, measured in tonnes per hour, was recorded every 2 hours. The results are shown in the table below.

For safety reasons oil rigs are required to shut down and stop all operations until an inspection is carried out should the total amount of oil leaked during any incident exceed 250 tonnes.

Use all the data in the table to decide if the AlphaBeta rig should be shut down or not.

Explain why using the trapezium rule to estimate the total amount of oil spilled from AlphaBeta in the first 10 hours would be an over-estimate.

A stretch of road along Equality Street has a speed limit of 70 mph and traffic is monitored by six average speed cameras.

Ricky is driving his car along Equality Street and passes the first camera at time zero.
Ricky's speed, measured in miles per hour, and the time he passes each camera, are recorded in the table below.

Use all the results above to estimate the distance between the first and last camera.

A driver will receive a speeding ticket if their average speed between the first and last camera exceeds the speed limit.

Should Ricky receive a speeding ticket or not? Justify your answer.

Use calculus to find the value of

${\int }_2^6\left(\frac{1}{\mathrm{x}}+\frac{4}{2\mathrm{x}}+\frac{3}{3\mathrm{x}-2}\right)\mathrm{dx}$

giving your answer in the form p ln p + q In q, where p and q are prime numbers to be found.

Find an expression for y given that

$\frac{\mathrm{dy}}{\mathrm{dx}}=5{\cos }^{2}4\mathrm{xsin}4\mathrm{x}$

Integrate

$\int 3\mathrm{x}{\left(5{\mathrm{x}}^2+4\right)}^4\mathrm{dx}$

Show that

$\left(\cos \left(\theta +\frac{\mathrm{\pi }}{8}\right)+\sin \left(\theta +\frac{\mathrm{\pi }}{8}\right)\right)\left(\sin \left(\theta +\frac{\mathrm{\pi }}{8}\right)-\cos \left(\theta +\frac{\mathrm{\pi }}{8}\right)\right)\equiv -\cos \left(2\theta +\frac{\mathrm{\pi }}{4}\right)$

Hence, or otherwise, find the exact value of

${\int }_0^{\frac{\mathrm{\pi }}{8}}\left({\sin }^2\left(\theta +\frac{\mathrm{\pi }}{8}\right)-{\cos }^2\left(\theta +\frac{\mathrm{\pi }}{8}\right)\right)\mathrm{d}\theta$

Show that

${\int }_0^1{\mathrm{e}}^{\mathrm{ax}+\mathrm{b}}\mathrm{dx}={\mathrm{e}}^{\mathrm{b}}\left(\frac{{\mathrm{e}}^{\mathrm{a}}-1}{\mathrm{a}}\right)$

where a and b are constants, and a # 0.

Using your working from part (a), or otherwise, evaluate

${\int }_0^{\mathrm{c}}{\mathrm{e}}^{\mathrm{ax}+\mathrm{b}}\mathrm{dx}$

giving your answer in terms of a, b and c, where a, b and c are constants, and a # 0.

Find

$\int {\left(2\mathrm{tanx}+3\mathrm{secx}\right)}^2\mathrm{dx}$

Show that

$\left(1+\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)\left(1-\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)\equiv 2-{\mathrm{cosec}}^2\left(2\theta +\frac{\mathrm{\pi }}{4}\right)$

Hence, or otherwise, find an expression for f($\theta$) given that

$\mathrm{f}\text{'}\left(\theta \right)=\left(2+2\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)\left(2-2\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)$

Show that there are no positive values of a and b that satisfy the equation

${\int }_0^1\frac{2\mathrm{a}}{\mathrm{ax}+\mathrm{b}}\mathrm{dx}={\int }_{\mathrm{b}}^{2\mathrm{a}}\frac{1}{\mathrm{x}}\mathrm{dx}$

Given that

$\mathrm{f}\text{'}\left(\theta \right)=\frac{9}{3-3{\sin }^2\left(3\theta -{\displaystyle \frac{\mathrm{\pi }}{6}}\right)}$

and also that

${\left[\mathrm{f}\left(\frac{\mathrm{\pi }}{6}\right)\right]}^2-{\left[\mathrm{f}\left(\frac{\mathrm{\pi }}{9}\right)\right]}^2=\frac{8}{3}\left(1-\sqrt{3}\right)$

find $\mathrm{f}\left(\theta \right)$

The trapezium rule is to be used to find an estimate for the integral

${\int }_4^8\mathrm{f}\left(x\right)\mathrm{dx}$

The table below shows values for x and f(x), rounded to three significant figures where appropriate.

Using the values in the table find

(i) an estimate for the integral using 2 strips,

(ii) an estimate for the integral using 4 strips,

(iii) an estimate for the integral using 8 strips.

Justify which of the estimates from part (a) will be the most accurate estimate for the integral.

The diagram below shows the graph of y = 4 - 2x^Inx, × > 0.

Use the trapezium rule with h = 0.2 to find an estimate of the integral

${\int }_1^2\left(4-2{\mathrm{x}}^{\mathrm{Inx}}\right)\mathrm{dx}$

to three significant figures.

Using the integration feature on your calculator, find the value of

${\int }_1^2\left(4-2{\mathrm{x}}^{\mathrm{Inx}}\right)\mathrm{dx}$

Give your answer to three significant figures.

Assuming your calculator provides the exact answer to the integral, find the percentage error of your estimate from part (a).

The diagram below shows a sketch of the graph of y = x(x-6)² x ≥0

The graph has a local maximum point at (2,32) as indicated on the diagram.

Use the trapezium rule with 5 ordinate values to estimate the area shaded.

Using the appropriate working values from part (a), find an upper and lower bound for the area shaded.

Suggest a reason why using the trapezium rule in this case is not appropriate.

The energy company Power² own and maintain a wind farm.

Engineers from Power² model the power output, P kW (kiloWatts), of a single wind turbine over a twelve-hour period according to the function

$\mathrm{p}\left(\mathrm{t}\right)=60+5{\mathrm{e}}^{0.7\sqrt{t}}\sin t$ 0≤ t ≤ 12

where t is the number of hours after 6pm. The graph of y = P(t) is shown below.

Estimate the total power generated by this wind turbine between midnight and 4am.

Once every twelve hours, Power? turn each wind turbine off for half an hour for maintenance and safety checks.

Suggest, with a reason, at what time between 6pm and 6am the maintenance and safety checks should be carried out on the wind turbine modelled above.

The XnValdez container ship was carrying 4000 tonnes of crude oil when it ran aground and oil started leaking from its hull into the ocean.

It took 36 days for experts to stop the oil leak.

During that time the leakage rate of the oil, measured in tonnes per day, was recorded every 4 days. The results are shown in the table below.

An environmental disaster is declared if the total amount of crude oil that leaks into the ocean exceeds 2500 tonnes.