| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Square of partial fractions expression |
| Difficulty | Standard +0.8 This question requires standard partial fractions (routine), then squaring the result and simplifying algebraically (moderately challenging manipulation), followed by integration of standard forms. The algebraic manipulation in part (ii) is non-trivial and requires careful expansion and collection of terms, elevating this above a typical textbook exercise. However, all techniques are standard for Further Pure Mathematics students. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply the form \(\frac{A}{x+1} + \frac{B}{x+3}\) and use a relevant method to find \(A\) or \(B\) | M1 | |
| Obtain \(A = 1\), \(B = -1\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Square the result of part (i) and substitute the fractions of part (i) | M1 | |
| Obtain the given answer correctly | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate and obtain \(-\frac{1}{x+1} - \ln(x+1) + \ln(x+3) - \frac{1}{x+3}\) | B3 | |
| Substitute limits correctly in an integral containing at least two terms of the correct form | M1 | |
| Obtain given answer following full and exact working | A1 | [5] |
## Question 8:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the form $\frac{A}{x+1} + \frac{B}{x+3}$ and use a relevant method to find $A$ or $B$ | M1 | |
| Obtain $A = 1$, $B = -1$ | A1 | [2] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Square the result of part (i) and substitute the fractions of part (i) | M1 | |
| Obtain the given answer correctly | A1 | [2] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate and obtain $-\frac{1}{x+1} - \ln(x+1) + \ln(x+3) - \frac{1}{x+3}$ | B3 | |
| Substitute limits correctly in an integral containing at least two terms of the correct form | M1 | |
| Obtain given answer following full and exact working | A1 | [5] |
---8 (i) Express $\frac { 2 } { ( x + 1 ) ( x + 3 ) }$ in partial fractions.\\
(ii) Using your answer to part (i), show that
$$\left( \frac { 2 } { ( x + 1 ) ( x + 3 ) } \right) ^ { 2 } \equiv \frac { 1 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x + 1 } + \frac { 1 } { x + 3 } + \frac { 1 } { ( x + 3 ) ^ { 2 } }$$
(iii) Hence show that $\int _ { 0 } ^ { 1 } \frac { 4 } { ( x + 1 ) ^ { 2 } ( x + 3 ) ^ { 2 } } \mathrm {~d} x = \frac { 7 } { 12 } - \ln \frac { 3 } { 2 }$.
\hfill \mbox{\textit{CAIE P3 2010 Q8 [9]}}