| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Substitution with exponential functions |
| Difficulty | Standard +0.3 Part (i) is a standard logarithmic integration where the numerator is the derivative of the denominator (requiring recognition of the pattern f'(x)/f(x)). Part (ii) is a routine substitution u = e^(2x) - 1, leading to a straightforward power rule integration. Both are textbook exercises testing recognition of standard techniques with minimal problem-solving required, making this slightly easier than average. |
| Spec | 1.08h Integration by substitution |
| VIIV SIHI NI JIIIM IONOO | VIUV SIHI NI III M M I ON OO | VI4V SIHI NI IIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\frac{3x-2}{3x^2-4x+5}\,dx = \frac{1}{2}\ln(3x^2-4x+5)(+c)\) | M1, A1 | M1: integrates to form \(\alpha\ln(3x^2-4x+5)\) where \(\alpha\) is a constant. Do not accept \(\alpha\ln(3x^2-4x+5)+f(x)\). A1: \(\frac{1}{2}\ln(3x^2-4x+5)\); bracket or modulus must be present. Do not penalise \(\frac{\ln(3x^2-4x+5)}{2}\) if intention is clear. Penalise spurious notation on A mark only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\frac{e^{2x}}{(e^{2x}-1)^3}\,dx = -\frac{1}{4}(e^{2x}-1)^{-2}(+c)\) | M1, A1 | M1: integrates to form \(\beta(e^{2x}-1)^{-2}\). Do not accept \(\beta(e^{2x}-1)^{-2}+g(x)\). Allow substitutions: \(u=e^{2x}-1\Rightarrow ku^{-2}\); \(u=e^x\Rightarrow k(u^2-1)^{-2}\); \(u=e^{2x}\Rightarrow k(u-1)^{-2}\). A1: \(-\frac{1}{4}(e^{2x}-1)^{-2}\) or exact equivalent with or without \(+c\). Penalise spurious notation on A mark only |
# Question 9:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\frac{3x-2}{3x^2-4x+5}\,dx = \frac{1}{2}\ln(3x^2-4x+5)(+c)$ | M1, A1 | M1: integrates to form $\alpha\ln(3x^2-4x+5)$ where $\alpha$ is a constant. Do not accept $\alpha\ln(3x^2-4x+5)+f(x)$. A1: $\frac{1}{2}\ln(3x^2-4x+5)$; bracket or modulus must be present. Do not penalise $\frac{\ln(3x^2-4x+5)}{2}$ if intention is clear. Penalise spurious notation on A mark only |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\frac{e^{2x}}{(e^{2x}-1)^3}\,dx = -\frac{1}{4}(e^{2x}-1)^{-2}(+c)$ | M1, A1 | M1: integrates to form $\beta(e^{2x}-1)^{-2}$. Do not accept $\beta(e^{2x}-1)^{-2}+g(x)$. Allow substitutions: $u=e^{2x}-1\Rightarrow ku^{-2}$; $u=e^x\Rightarrow k(u^2-1)^{-2}$; $u=e^{2x}\Rightarrow k(u-1)^{-2}$. A1: $-\frac{1}{4}(e^{2x}-1)^{-2}$ or exact equivalent with or without $+c$. Penalise spurious notation on A mark only |
---9. Find\\
(i) $\int \frac { 3 x - 2 } { 3 x ^ { 2 } - 4 x + 5 } \mathrm {~d} x$\\
(ii) $\int \frac { \mathrm { e } ^ { 2 x } } { \left( \mathrm { e } ^ { 2 x } - 1 \right) ^ { 3 } } \mathrm {~d} x \quad x \neq 0$
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VIIV SIHI NI JIIIM IONOO & VIUV SIHI NI III M M I ON OO & VI4V SIHI NI IIIYM ION OC \\
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\hfill \mbox{\textit{Edexcel P3 2021 Q9 [4]}}