Question 4 - A-Level Maths

Edexcel P4 2021 June — Question 4 8 marks

Exam Board	Edexcel
Module	P4 (Pure Mathematics 4)
Year	2021
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration by Substitution
Type	Partial fractions after substitution
Difficulty	Standard +0.8 This P4 question requires multiple sophisticated techniques: applying the given substitution u=√x (including finding du and changing limits), factoring the resulting denominator, decomposing into partial fractions, integrating logarithmic terms, and simplifying to the exact form. While each step is standard for Further Maths, the combination of substitution + partial fractions + exact answer manipulation makes this moderately challenging, above average difficulty but not exceptional for P4 level.
Spec	1.08h Integration by substitution

4. Use algebraic integration and the substitution $u = \sqrt { x }$ to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 10 } { 5 x + 2 x \sqrt { x } } \mathrm {~d} x$$ Write your answer in the form $4 \ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers to be found.
(Solutions relying entirely on calculator technology are not acceptable.)

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Question 4:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$u = \sqrt{x} \Rightarrow x = u^2 \Rightarrow \frac{dx}{du} = 2u$	B1	For $\frac{dx}{du}=2u$ or $\frac{du}{dx}=\frac{1}{2}x^{-\frac{1}{2}}$ or equivalent
$\int\frac{10}{5x+2x\sqrt{x}}\,dx = \int\frac{10}{5u^2+2u^3}\cdot 2u\,du$	M1 A1	M1: Attempts to write all terms in $u$ (inc dx). dx CANNOT just be replaced by du. Look for $x\to u^2$ or $x\sqrt{x}\to u^3$ with $dx\to f(u)du$. A1: Correct integrand in terms of $u$, may be unsimplified
$= \int\frac{20}{5u+2u^2}\,du = \int\frac{4}{u} - \frac{8}{5+2u}\,du$	dM1 A1	dM1: Attempts partial fractions, writing integrand as component fractions. A1: Correct PF $\int\frac{4}{u}-\frac{8}{5+2u}\,(du)$
$= 4\ln u - 4\ln(5+2u)$	ddM1	For $...\ln u \pm ...\ln(5+2u)$, following through on their PFs
$\left[4\ln u - 4\ln(5+2u)\right]_1^2 = 4\ln 2 - 4\ln 9 + 4\ln 7 = ...$	M1	Uses limits 1 and 2 (or substitutes $u=\sqrt{x}$ and uses limits 1 and 4)
$= 4\ln\left(\frac{14}{9}\right)$	A1

# Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $u = \sqrt{x} \Rightarrow x = u^2 \Rightarrow \frac{dx}{du} = 2u$ | B1 | For $\frac{dx}{du}=2u$ or $\frac{du}{dx}=\frac{1}{2}x^{-\frac{1}{2}}$ or equivalent |
| $\int\frac{10}{5x+2x\sqrt{x}}\,dx = \int\frac{10}{5u^2+2u^3}\cdot 2u\,du$ | M1 A1 | M1: Attempts to write all terms in $u$ (inc dx). dx CANNOT just be replaced by du. Look for $x\to u^2$ or $x\sqrt{x}\to u^3$ with $dx\to f(u)du$. A1: Correct integrand in terms of $u$, may be unsimplified |
| $= \int\frac{20}{5u+2u^2}\,du = \int\frac{4}{u} - \frac{8}{5+2u}\,du$ | dM1 A1 | dM1: Attempts partial fractions, writing integrand as component fractions. A1: Correct PF $\int\frac{4}{u}-\frac{8}{5+2u}\,(du)$ |
| $= 4\ln u - 4\ln(5+2u)$ | ddM1 | For $...\ln u \pm ...\ln(5+2u)$, following through on their PFs |
| $\left[4\ln u - 4\ln(5+2u)\right]_1^2 = 4\ln 2 - 4\ln 9 + 4\ln 7 = ...$ | M1 | Uses limits 1 and 2 (or substitutes $u=\sqrt{x}$ and uses limits 1 and 4) |
| $= 4\ln\left(\frac{14}{9}\right)$ | A1 | |

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4. Use algebraic integration and the substitution $u = \sqrt { x }$ to find the exact value of

$$\int _ { 1 } ^ { 4 } \frac { 10 } { 5 x + 2 x \sqrt { x } } \mathrm {~d} x$$

Write your answer in the form $4 \ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers to be found.\\
(Solutions relying entirely on calculator technology are not acceptable.)\\

\hfill \mbox{\textit{Edexcel P4 2021 Q4 [8]}}

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