CAIE A-Level Mathematics AS 2.5.1 Extended Integration Rules Questions

Practise integrating exponential, reciprocal-linear and trigonometric forms by reverse differentiation, retaining scale factors and evaluating exact definite integrals or areas.

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[Maximum number: 5]

Figure for Question CAIE A-Level Mathematics AS 2.5.1 Extended Integration Rules Questions question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

Show that the x-coordinate of B is $\ln 2$ and hence find the area of the shaded region.

Attempt to find x-coordinate of B
$8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}=0$ .

Obtain $\mathrm{e}^{3 x}=8$ and hence $x=\ln 2$
AG so necessary detail needed.

A0 if decimals used.

Integrate to obtain $-8 \mathrm{e}^{-x}-\frac{1}{2} \mathrm{e}^{2 x}$
Use limits 0 and $\ln 2$ correctly to find area
For integral of form $k_{3} \mathrm{e}^{-x}+k_{4} \mathrm{e}^{2 x}$ where $k_{3} k_{4} \neq 0$ .
$k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $\frac{5}{2}$
OE

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CAIE A-Level Mathematics AS 2.5.1 Extended Integration Rules Questions question 1