## Question 11(a):

| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\dfrac{Ax+B}{4+x^2}+\dfrac{C}{1+x}$ | **B1** | |
| Use a correct method for finding a coefficient | **M1** | $(Ax+B)(1+x)+C(4+x^2)=5x^2+x+11$ |
| Obtain one of $A=2$, $B=-1$ and $C=3$ | **A1** | If error present in above still allow A1 for $C$ |
| Obtain a second value | **A1** | |
| Obtain the third value | **A1** | If $A=0$ then max M1 A1 (for $C$) |
| | **5** | |

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## Question 11(b):

| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate and obtain terms $\left(\dfrac{A}{2}\right)\ln(4+x^2)+\dfrac{B}{2}\tan^{-1}\!\left(\dfrac{x}{2}\right)+C\ln(1+x)$ | **B1FT + B1FT + B1FT** | FT is on $A$, $B$ and $C$. Integral of $\dfrac{Ax+B}{4+x^2}=\dfrac{B}{2}\tan^{-1}\!\left(\dfrac{x}{2}\right)$ or $(A/2)\ln(4+x^2)$ only. B0FT unless clearly from single term. |
| Substitute limits 0 and 2 correctly in an integral of the form $a\ln(4+x^2)+b\tan^{-1}\!\left(\dfrac{x}{2}\right)+c\ln(1+x)$, where $abc\neq 0$ | **M1** | $a\ln(4+4)+b\tan^{-1}\!\left(\dfrac{2}{2}\right)+c\ln(1+2)-\left[a\ln 4+b\tan^{-1}0+c\ln(1)\right]$. Allow one sign or substitution error. Allow omission of $b\tan^{-1}0+c\ln(1)$. |
| Obtain answer $\ln 54-\dfrac{\pi}{8}$ after full and correct working | **A1** | AG – work to combine or simplify at least 2 of ln terms is required CWO. A0 if any non exact value(s) seen. |
| | **5** | |