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A-Level CAIE Mathematics AS2.5 IntegrationQuestion Bank

2.5 Integration

Question 1

A curve is such that dy dx=47−2x\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{4}{7-2 x}. The point (3,2) lies on the curve. Find the equation of the curve.

Question 1

[Maximum number: 3]

Use the trapezium rule with 3 intervals to estimate the value of

∫03∣2x−4∣dx.\int_{0}^{3}\left|2^{x}-4\right| \mathrm{d} x .

Question 1

[Maximum number: 3]

Use the trapezium rule with three intervals to estimate the value of

∫014π(1−tan⁡x)dx\int_{0}^{\frac{1}{4} \pi} \sqrt{ }(1-\tan x) \mathrm{d} x

giving your answer correct to 3 decimal places.

Question 1

[Maximum number: 3]

Use the trapezium rule with four intervals to find an approximation to

∫15∣2x−8∣dx\int_{1}^{5}\left|2^{x}-8\right| \mathrm{d} x

Question 1

[Maximum number: 4]
Question image

The diagram shows a sketch of the curve y=3(9−x3)y=\frac{3}{\sqrt{ }\left(9-x^{3}\right)} for values of x from -1.2 to 1.2 .

Question 1(i)

(a)

Use the trapezium rule, with two intervals, to estimate the value of

∫−1.21.23(9−x3)dx\int_{-1.2}^{1.2} \frac{3}{\sqrt{ }\left(9-x^{3}\right)} \mathrm{d} x

giving your answer correct to 2 decimal places.

[ 3 ]

Question 1(ii)

(b)

Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.

[ 1 ]

Question 1

[Maximum number: 3]

Use the trapezium rule with three intervals to estimate the value of

∫012πln⁡(1+sin⁡x)dx\int_{0}^{\frac{1}{2} \pi} \ln (1+\sin x) \mathrm{d} x

giving your answer correct to 2 decimal places.

Question 1

Question 1(i)

(a)

Find ∫24x−1 dx\int \frac{2}{4 x-1} \mathrm{~d} x.

Question 1(ii)

(b)

Hence find ∫1724x−1 dx\int_{1}^{7} \frac{2}{4 x-1} \mathrm{~d} x, expressing your answer in the form ln⁡a\ln a, where a is an integer.

[ 3 ]

Question 2

[Maximum number: 3]

Show that ∫2624x+1 dx=ln⁡53\int_{2}^{6} \frac{2}{4 x+1} \mathrm{~d} x=\ln \frac{5}{3}.

Question 2

[Maximum number: 4]
Question image

The diagram shows part of the curve y=xe−xy=x \mathrm{e}^{-x}. The shaded region R is bounded by the curve and by the lines x=2, x=3 and y=0.

Question 2(i)

(a)

Use the trapezium rule with two intervals to estimate the area of R, giving your answer correct to 2 decimal places.

[ 3 ]

Question 2(ii)

(b)

State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of R.

[ 1 ]

Question 2

Use the trapezium rule with three intervals to find an approximation to

∫03∣3x−10∣dx\int_{0}^{3}\left|3^{x}-10\right| \mathrm{d} x
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