| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.8 This FP2 question requires partial fractions with a quartic denominator factorization (recognizing 1-x⁴ = (1-x)(1+x)(1+x²)), then integrating terms including arctan for the irreducible quadratic. The algebraic manipulation is substantial and the definite integral evaluation requires careful handling of logarithms and inverse trig functions to reach the specific form required. More demanding than typical C3/C4 integration but standard for FP2. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt at correct form of P.F. | M1 | Allow \(\frac{Cx}{(x^2+1)}\) here; not \(C = 0\) |
| Rewrite as \(A(1+x)(1+x^2) + B(1-x)(1+x^2) + (Cx+D)(1-x)(1+x)\) | M1 | From their P.F. |
| Use values of x/equate coefficients | M1 | |
| Get \(A = 1, B = 1\) | A1 | cwo |
| Get \(C = 0, D = 2\) | A1 | |
| SC Use of cover-up rule for A,B | M1 | |
| If both correct | A1 | cwo |
| Answer | Marks | Guidance |
|---|---|---|
| Get \(A\ln(1+x) - B\ln(1-x)\) | M1 | Or quote from List of Formulae |
| Get \(\frac{2\tan^{-1}x}{1}\) | B1 | |
| Use limits in their integrated expressions | M1 | |
| Clearly get A.G. | A1 |
## (i)
Attempt at correct form of P.F. | M1 | Allow $\frac{Cx}{(x^2+1)}$ here; not $C = 0$
Rewrite as $A(1+x)(1+x^2) + B(1-x)(1+x^2) + (Cx+D)(1-x)(1+x)$ | M1 | From their P.F.
Use values of x/equate coefficients | M1 |
Get $A = 1, B = 1$ | A1 | cwo
Get $C = 0, D = 2$ | A1 |
**SC** Use of cover-up rule for A,B | M1 |
If both correct | A1 | cwo
## (ii)
Get $A\ln(1+x) - B\ln(1-x)$ | M1 | Or quote from List of Formulae
Get $\frac{2\tan^{-1}x}{1}$ | B1 |
Use limits in their integrated expressions | M1 |
Clearly get A.G. | A1 |
---\begin{enumerate}[label=(\roman*)]
\item Express $\frac{4}{(1-x)(1+x)(1+x^2)}$ in partial fractions. [5]
\item Show that $\int_0^{\frac{\sqrt{3}}{3}} \frac{4}{1-x^4} dx = \ln\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) + \frac{1}{3}\pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q6 [9]}}