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Date	May Example questions	Marks available	6	Reference code	EXM.1.AHL.TZ0.2
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Express	Question number	2	Adapted from	N/A

Question

Consider the integral $\int\limits_1^t {\frac{{ - 1}}{{x + {x^2}}}{\text{ }}} dx$ for $t > 1$ .

Very briefly, explain why the value of this integral must be negative.

[1]

Express the function $f\left( x \right) = \frac{{ - 1}}{{x + {x^2}}}$ in partial fractions.

[6]

Use parts (a) and (b) to show that ${\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t < {\text{ln}}\,2$ .

[4]

Markscheme

The numerator is negative but the denominator is positive. Thus the integrand is negative and so the value of the integral will be negative. R1AG

[1 mark]

$\frac{{ - 1}}{{x + {x^2}}} = \frac{{ - 1}}{{\left( {1 + x} \right)x}} \equiv \frac{A}{{1 + x}} + \frac{B}{x}$ M1M1A1

$\Rightarrow - 1 \equiv Ax + B(1 + x) \Rightarrow A = 1,\,B = - 1$ M1A1

$\frac{{ - 1}}{{x + {x^2}}} \equiv \frac{1}{{1 + x}} + \frac{{ - 1}}{x}$ A1

[6 marks]

$\int\limits_1^t {\frac{1}{{1 + x}} + \frac{{ - 1}}{x}dx} = \left[ {{\text{ln}}\left( {1 + x} \right) - {\text{ln}}\,x} \right]_1^t = {\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t - {\text{ln}}\,2$ M1A1A1

Hence ${\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t - {\text{ln}}\,2 < 0 \Rightarrow {\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t < {\text{ln}}\,2$ R1AG

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » AHL 1.11—Partial fractions

Show 42 related questions

22M.2.AHL.TZ2.12e:
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21N.2.AHL.TZ0.10d:
Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.
EXM.1.AHL.TZ0.2a:
Very briefly, explain why the value of this integral must be negative.
EXM.3.AHL.TZ0.4a:
Show that the general solution of this differential equation is $P = A{{\text{e}}^{kt}}$ , where $A \in \mathbb{R}$ .
EXM.3.AHL.TZ0.4b.iii:
$\mathop {{\text{lim}}}\limits_{t \to \infty } P$
EXM.3.AHL.TZ0.4c.i:
the solution of the differential equation, giving your answer in the form $P = f\left( t \right)$ .
EXM.3.AHL.TZ0.4f:
Given that the initial population is 1000, $L = 10000$ and $m = 0.003$ , find the number of years it will take for the population to triple.
EXM.3.AHL.TZ0.4d:
Show that $\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)$ , where $m \in \mathbb{R}$ .
EXM.3.AHL.TZ0.4b.ii:
the number of years it will take for the population to triple.
EXM.1.AHL.TZ0.3b:
Use part (a) to show that $f(x)$ is always decreasing.
22M.2.AHL.TZ1.12c.i:
By solving the differential equation, show that $y = \frac{8 x + x^{4}}{4 - x^{3}}$ .
22M.2.AHL.TZ1.12c.ii:
Find the actual value of $y$ when $x = 1.5$ .
20N.1.AHL.TZ0.H_12b:
State the equation of the horizontal asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12d:
Sketch the graph of $y = f (x)$ , stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.
20N.2.AHL.TZ0.H_3b:
Hence find the area of the shaded region.
EXM.3.AHL.TZ0.4c.ii:
the number of years it will take for the population to triple.
EXM.1.AHL.TZ0.3a:
Express $f(x)$ in partial fractions.
EXN.2.AHL.TZ0.7c:
Give a reason why the series expansion found in part (b) is not valid for $x = \frac{3}{4}$ .
21M.2.AHL.TZ1.12a:
The expression for $f' (x)$ can be written in the form $\frac{a}{x} + \frac{b}{k - x}$ , where $a, b \in ℝ$ . Find $a$ and $b$ in terms of $k$ .
EXM.3.AHL.TZ0.4b.i:
the population after 10 years
20N.2.AHL.TZ0.H_3a:
Determine the values of $p$ , $q$ and $r$ .
19N.2.AHL.TZ0.H_11c:
The region $R$ is now rotated about the $y$ -axis, through $2\pi$ radians, to form a solid.

By writing ${{\text{si}}{{\text{n}}^3}\,y}$ as $\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y$ , show that the volume of the solid formed is $\frac{{2\pi }}{3}$ .
18N.2.AHL.TZ0.H_1b:
Find the sum to infinity of this sequence.
21M.2.AHL.TZ1.12b:
Hence, find an expression for $f (x)$ .
21N.2.AHL.TZ0.10a.ii:
$y$ -axis.
21N.2.AHL.TZ0.10c:
The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .
EXM.1.AHL.TZ0.2c:
Use parts (a) and (b) to show that ${\text{ln}}\left( {1 + t} \right) - {\text{ln}}\,t < {\text{ln}}\,2$ .
EXN.2.AHL.TZ0.7a:
Find the value of $A$ and the value of $B$ .
20N.1.AHL.TZ0.H_12a:
State the equation of the vertical asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12c:
Use an algebraic method to determine whether $f$ is a self-inverse function.
19N.1.AHL.TZ0.H_10c:
Show that $\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}$ .
EXM.3.AHL.TZ0.4e:
Solve the differential equation $\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)$ , giving your answer in the form $P = g\left( t \right)$ .
20N.1.AHL.TZ0.H_12e:
The region bounded by the $x$ -axis, the curve $y = f (x)$ , and the lines $x = 5$ and $x = 7$ is rotated through $2 π$ about the $x$ -axis. Find the volume of the solid generated, giving your answer in the form $π (a + b \ln 2)$ , where $a, b \in ℤ$ .
19N.2.AHL.TZ0.H_11b:
Find the area of $R$ .
EXM.1.AHL.TZ0.1:
Let $f(x) = \frac{1}{{1 - {x^2}}}$ for $- 1 < x < 1$ . Use partial fractions to find $\int {f\left( x \right)} {\text{ }}dx$ .
EXM.1.AHL.TZ0.3c:
Use part (a) to find the exact value of $\int\limits_{ - 1}^0 {f(x)dx}$ , giving the answer in the form ${\text{ln}}\,q$ , $q \in \mathbb{Q}$ .
19N.1.AHL.TZ0.H_10d:
The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .
EXN.2.AHL.TZ0.7b:
Hence, expand $\frac{2 + 7 x}{(1 + 2 x) (1 - x)}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .
21N.2.AHL.TZ0.10e.i:
Express $\frac{1}{f (x)}$ in partial fractions.
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21N.2.AHL.TZ0.10a.i:
$x$ -axis.
21N.2.AHL.TZ0.10b:
Write down the equation of the vertical asymptote of the graph of $f$ .

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