[Maximum number: 3]

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

\int_{1}^{5}\left|2^{x}-8\right| \mathrm{d} x

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

(a)

Find $\int \frac{2}{4 x-1} \mathrm{~d} x$ .

(b)

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

[ 3 ]

[Maximum number: 4]

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

(a)

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

\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 ]

(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 ]

[Maximum number: 3]

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

\int_{0}^{\frac{1}{2} \pi} \ln (1+\sin x) \mathrm{d} x

giving your answer correct to 2 decimal places.

[Maximum number: 3]

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

\int_{0}^{\frac{1}{4} \pi} \sqrt{ }(1-\tan x) \mathrm{d} x

giving your answer correct to 3 decimal places.

[Maximum number: 3]

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

\int_{0}^{3}\left|2^{x}-4\right| \mathrm{d} x .

(a)

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

\int_{\frac{1}{6} \pi}^{\frac{2}{3} \pi} \operatorname{cosec} x \mathrm{~d} x

giving your answer correct to 2 decimal places.

[ 3 ]

(b)

Using a sketch of the graph of $y=\operatorname{cosec} x$ , explain whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).

[ 2 ]

[Maximum number: 3]

Show that $\int_{2}^{6} \frac{2}{4 x+1} \mathrm{~d} x=\ln \frac{5}{3}$ .

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

\int_{0}^{3}\left|3^{x}-10\right| \mathrm{d} x