CAIE A-Level Mathematics 2.5.1 Extended Integration Rules

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

Exam points

integrate e^(ax+b) by dividing by a and integrate 1/(ax+b) as (1/a)ln|ax+b|
integrate sin, cos or sec² of a linear argument with the correct reciprocal scale factor
apply exact bounds and use logarithm laws or intersections to simplify the final value or area

[Maximum number: 5]

Figure for Question 2.5.1—Extended integration rules 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

M1

$8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}=0$ .

Obtain $\mathrm{e}^{3 x}=8$ and hence $x=\ln 2$

A1

AG so necessary detail needed.

A0 if decimals used.

Integrate to obtain $-8 \mathrm{e}^{-x}-\frac{1}{2} \mathrm{e}^{2 x}$

B1

Use limits 0 and $\ln 2$ correctly to find area

M1

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 $\frac{5}{2}$

A1

OE

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2.5.1—Extended integration rules question 1