Question 10 - A-Level Maths

CAIE P3 2014 November — Question 10 10 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2014
Session	November
Marks	10
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration by Substitution
Type	Show definite integral equals specific value (requiring partial fractions or complex algebra)
Difficulty	Standard +0.3 This is a straightforward substitution question where the substitution is given explicitly. After substituting u = e^x, the integral becomes a rational function requiring partial fractions (a standard P3 technique), followed by routine integration of logarithmic terms and evaluation at transformed limits. While it requires multiple steps, each is a standard textbook procedure with no novel insight needed, making it slightly easier than average.
Spec	1.02y Partial fractions: decompose rational functions 1.08h Integration by substitution

10 By first using the substitution $u = \mathrm { e } ^ { x }$, show that $$\int _ { 0 } ^ { \ln 4 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } + 3 \mathrm { e } ^ { x } + 2 } \mathrm {~d} x = \ln \left( \frac { 8 } { 5 } \right)$$

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Answer	Marks	Guidance
State or imply $\frac{du}{dx} = e^x$	B1
Substitute throughout for $x$ and $dx$	M1
Obtain $\int\frac{u}{u^2 + 3u + 2}du$ or equivalent (ignoring limits so far)	A1
State or imply partial fractions of form $\frac{A}{u+2} + \frac{B}{u+1}$, following their integrand	B1
Carry out a correct process to find at least one constant for their integrand	M1
Obtain correct $\frac{2}{u+2} - \frac{1}{u+1}$	A1
Integrate to obtain $a\ln(u+2) + b\ln(u+1)$	M1
Obtain $2\ln(u+2) - \ln(u+1)$ or equivalent, follow their $A$ and $B$	A1♦
Apply appropriate limits and use at least one logarithm property correctly	M1
Obtain given answer $\ln\frac{5}{3}$ legitimately	A1	[10]
SR for integrand $\frac{u^2}{u(u+1)(u+2)}$
State or imply partial fractions of form $\frac{A}{u} + \frac{B}{u+1} + \frac{C}{u+2}$	(B1)
Carry out a correct process to find at least one constant	(M1)
Obtain correct $\frac{2}{u+2} - \frac{1}{u+1}$	(A1)
$\ldots$complete as above.

State or imply $\frac{du}{dx} = e^x$ | B1 |
Substitute throughout for $x$ and $dx$ | M1 |
Obtain $\int\frac{u}{u^2 + 3u + 2}du$ or equivalent (ignoring limits so far) | A1 |
State or imply partial fractions of form $\frac{A}{u+2} + \frac{B}{u+1}$, following their integrand | B1 |
Carry out a correct process to find at least one constant for their integrand | M1 |
Obtain correct $\frac{2}{u+2} - \frac{1}{u+1}$ | A1 |
Integrate to obtain $a\ln(u+2) + b\ln(u+1)$ | M1 |
Obtain $2\ln(u+2) - \ln(u+1)$ or equivalent, follow their $A$ and $B$ | A1♦ |
Apply appropriate limits and use at least one logarithm property correctly | M1 |
Obtain given answer $\ln\frac{5}{3}$ legitimately | A1 | [10]

**SR for integrand** $\frac{u^2}{u(u+1)(u+2)}$ |  |
State or imply partial fractions of form $\frac{A}{u} + \frac{B}{u+1} + \frac{C}{u+2}$ | (B1) |
Carry out a correct process to find at least one constant | (M1) |
Obtain correct $\frac{2}{u+2} - \frac{1}{u+1}$ | (A1) |
$\ldots$complete as above. |  |

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10 By first using the substitution $u = \mathrm { e } ^ { x }$, show that

$$\int _ { 0 } ^ { \ln 4 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } + 3 \mathrm { e } ^ { x } + 2 } \mathrm {~d} x = \ln \left( \frac { 8 } { 5 } \right)$$

\hfill \mbox{\textit{CAIE P3 2014 Q10 [10]}}

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Q1 4 Q2 5 Q3 7 Q4 7 Q5 7 Q6 8 Q7 8 Q8 9 Q9 10 Q10 10

Answer	Marks	Guidance
State or imply \(\frac{du}{dx} = e^x\)	B1
Substitute throughout for \(x\) and \(dx\)	M1
Obtain \(\int\frac{u}{u^2 + 3u + 2}du\) or equivalent (ignoring limits so far)	A1
State or imply partial fractions of form \(\frac{A}{u+2} + \frac{B}{u+1}\), following their integrand	B1
Carry out a correct process to find at least one constant for their integrand	M1
Obtain correct \(\frac{2}{u+2} - \frac{1}{u+1}\)	A1
Integrate to obtain \(a\ln(u+2) + b\ln(u+1)\)	M1
Obtain \(2\ln(u+2) - \ln(u+1)\) or equivalent, follow their \(A\) and \(B\)	A1♦
Apply appropriate limits and use at least one logarithm property correctly	M1
Obtain given answer \(\ln\frac{5}{3}\) legitimately	A1	[10]
SR for integrand \(\frac{u^2}{u(u+1)(u+2)}\)
State or imply partial fractions of form \(\frac{A}{u} + \frac{B}{u+1} + \frac{C}{u+2}\)	(B1)
Carry out a correct process to find at least one constant	(M1)
Obtain correct \(\frac{2}{u+2} - \frac{1}{u+1}\)	(A1)
\(\ldots\)complete as above.