Integration techniques and differential equations – Year 13 A level maths worksheet with answers to download as PDF

Find each integral, or evaluate it where limits are given.

1. \(\int e^{3x}\,dx\) [1 mark]
2. \(\int\dfrac{1}{2x+5}\,dx\) [2 marks]
3. \(\int\sin4x\,dx\) [2 marks]
4. \(\int_0^{\pi/2}\sin2x\,dx\) [3 marks]

Use the reverse chain rule \(\int\dfrac{f\prime(x)}{f(x)}dx=\ln|f(x)|+c\) to find:

1. \(\int\dfrac{2x}{x^2+3}\,dx\) [2 marks]
2. \(\int\tan x\,dx\) [3 marks]
3. \(\int_0^1\dfrac{e^x}{e^x+1}\,dx\), giving an exact answer [3 marks]

Use a suitable substitution to evaluate each integral.

1. \(\int_0^2x(x^2+1)^3\,dx\) using \(u=x^2+1\) [4 marks]
2. \(\int_0^1xe^{x^2}\,dx\) using \(u=x^2\) [4 marks]

Use the substitution \(u=x+1\) to evaluate \(\int_0^3x\sqrt{x+1}\,dx\).

Use integration by parts to find or evaluate:

1. \(\int xe^{2x}\,dx\) [4 marks]
2. \(\int_1^ex\ln x\,dx\) [5 marks]

Express \(\dfrac{7x+1}{(x-1)(x+3)}\) in partial fractions and hence evaluate \(\int_2^4\dfrac{7x+1}{(x-1)(x+3)}\,dx\), giving your answer in the form \(a\ln3+b\ln\dfrac75\).

Water flows into a tank at a rate of \(R=12e^{-0.2t}\) litres per minute, where \(t\) is the time in minutes after a tap is opened.

1. Find the rate when \(t=0\) and when \(t=10\). [2 marks]
2. Find the volume of water that flows in during the first 10 minutes. [3 marks]
3. Show that the volume flowing in can never exceed 60 litres. [3 marks]

The speed of a sprinter during the first 4 seconds of a race is modelled by \(v=3te^{-0.5t}\) m s\(^{-1}\), where \(t\) is the time in seconds.

1. Find the time at which the speed is greatest and state this speed. [4 marks]
2. Use integration by parts to find the distance run in the first 4 seconds. [5 marks]
3. Find the average speed over the first 4 seconds. [1 mark]

The temperature \(\theta\) degrees Celsius of a cup of coffee in a room at \(20^\circ\)C satisfies the law of cooling \(\dfrac{d\theta}{dt}=-k(\theta-20)\), where \(t\) is in minutes and \(k>0\). Initially \(\theta=90\) and after 10 minutes \(\theta=55\).

1. Solve the differential equation to show that \(\theta=20+70e^{-kt}\). [4 marks]
2. Find the exact value of \(k\). [3 marks]
3. The coffee is drinkable below \(30^\circ\)C. Find when this happens. [3 marks]

A herd of deer in a national park has population \(P\) at time \(t\) years, with \(\dfrac{dP}{dt}=0.04P\). Initially there are 500 deer.

1. Solve the differential equation. [4 marks]
2. Find how long it takes the herd to double. [3 marks]
3. Give two reasons why the model may become unrealistic. [2 marks]

Water drains from a cylindrical tank so that the depth \(h\) metres at time \(t\) minutes satisfies \(\dfrac{dh}{dt}=-k\sqrt h\) for a positive constant \(k\). At \(t=0\), \(h=9\), and at \(t=5\), \(h=4\).

1. Solve the differential equation to show that \(2\sqrt h=6-kt\). [4 marks]
2. Find \(k\). [2 marks]
3. Find when the tank is empty. [2 marks]
4. Explain why the formula is not valid after this time. [1 mark]

A theatre sells tickets for a new show at a rate of \(S(t)=40te^{-0.2t}\) tickets per day, where \(t\) is the number of days since sales opened.

1. Find the exact number of tickets sold in the first 10 days. [5 marks]
2. The theatre holds 1200 seats. Show that the show can never sell out. [3 marks]
3. Find the day on which the rate of sales is greatest. [2 marks]

The power in an electrical circuit is \(P=P_0\sin^2t\), where \(t\) is the time in seconds and \(P_0\) is a constant in watts.

1. Use the identity \(\sin^2t=\dfrac{1-\cos2t}2\) to find \(\int_0^\pi\sin^2t\,dt\). [4 marks]
2. Hence find the mean value of \(\sin^2t\) over one half-cycle \(0\le t\le\pi\) and the mean power. [3 marks]

A factory discharges a pollutant into a lake. The mass \(m\) kg of pollutant in the lake after \(t\) days satisfies \(\dfrac{dm}{dt}=0.5-0.02m\), with \(m=0\) when \(t=0\).

1. Show that \(m=25\left(1-e^{-0.02t}\right)\). [5 marks]
2. Find the long-term mass of pollutant. [1 mark]
3. Find when the mass reaches 20 kg. [3 marks]

The diagram shows the curve \(y=xe^{-x}\). The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x=3\).

1. Find the coordinates of the maximum point of the curve. [4 marks]
2. Use integration by parts to show that the area of \(R\) is \(1-4e^{-3}\). [5 marks]
3. Evaluate the area to 3 significant figures. [1 mark]
4. Use the trapezium rule with 3 strips to estimate the area and comment on the accuracy. [3 marks]

A rectangular tank of base 4 m\(^2\) is filled with water to a depth that varies as the tank is lowered into a pool. The area of a horizontal cross-section at height \(x\) metres is \(A(x)=x^2\) square metres for \(0\le x\le3\).

1. Explain, by dividing the height into thin slices of thickness \(\delta x\), why the volume of the solid is \(\lim_{\delta x\to0}\sum_{x=0}^{3}x^2\delta x\). [3 marks]
2. Evaluate this limit. [2 marks]
3. The tank is filled by a pump at 0.5 m\(^3\) per minute. Find the time to fill the volume. [2 marks]
4. Use a sum of 3 slices of thickness 1 m (using the left-hand ends) to estimate the volume, and explain why it is an underestimate. [3 marks]

A population \(P\) (in thousands) of fish in a lake satisfies \(\dfrac{dP}{dt}=0.02P(5-P)\), where \(t\) is in years. Initially \(P=1\).

1. Show that \(\dfrac{1}{P(5-P)}=\dfrac15\left(\dfrac1P+\dfrac1{5-P}\right)\). [2 marks]
2. Solve the differential equation to show that \(P=\dfrac{5e^{0.1t}}{4+e^{0.1t}}\). [7 marks]
3. Find when the population reaches 4000. [3 marks]
4. State the long-term population and explain the shape of the model. [2 marks]

A footbridge arch has height \(y=\dfrac{x}{\sqrt{4-x^2}}\) metres above a horizontal line at a horizontal distance \(x\) metres, for \(0\le x\le1.5\) in a design drawing. Part of the drawing is used for a stained glass panel.

1. Use the substitution \(u=4-x^2\) to find \(\int\dfrac{x}{\sqrt{4-x^2}}dx\). [4 marks]
2. Find the area under the curve between \(x=0\) and \(x=1.5\). [3 marks]
3. Show that the area between \(x=0\) and \(x=2\) would be exactly 2 m\(^2\), and explain why the integral up to 2 needs care. [3 marks]

The current in a circuit after a switch is closed is \(I=t^2e^{-t}\) amperes, where \(t\) is in seconds. The charge that has flowed is \(Q=\int_0^TI\,dt\) coulombs.

1. Use integration by parts twice to show that \(\int t^2e^{-t}dt=-e^{-t}\left(t^2+2t+2\right)+c\). [6 marks]
2. Find the charge in the first 3 seconds. [2 marks]
3. Find the total charge as \(T\to\infty\) and comment. [2 marks]

Let \(J=\int_0^{\pi/2}e^x\sin x\,dx\).

1. Use integration by parts twice to show that \(2J=e^{\pi/2}+1\). [6 marks]
2. Find the value of \(J\) to 3 significant figures. [1 mark]

Let \(K=\int_0^{\pi/2}\dfrac{\sin x}{\sin x+\cos x}dx\) and \(L=\int_0^{\pi/2}\dfrac{\cos x}{\sin x+\cos x}dx\).

1. Show that \(K+L=\dfrac\pi2\). [2 marks]
2. Use the substitution \(u=\dfrac\pi2-x\) to show that \(K=L\). [4 marks]
3. Hence find the exact value of \(K\). [1 mark]