## Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| Split fraction to obtain $1 + \dfrac{x-4}{x^2+4}$ | B1 | |
| Attempt integration and obtain $p\ln(x^2+4)$ or $q\tan^{-1}\left(\dfrac{x}{2}\right)$ from correct working | M1 | Allow for $p\ln(x^2+4)$ from $\int \dfrac{x}{x^2+4}dx$ but only if a correct method for splitting has been used |
| Obtain $\dfrac{1}{2}\ln(x^2+4)$ | A1 FT | Follow through on their coefficients in the partial fraction. Allow from $\dfrac{x^2}{x^2+4} + \dfrac{x}{x^2+4}$ even if split is incomplete. Only available from a correct split. |
| Obtain $-2\tan^{-1}\left(\dfrac{x}{2}\right)$ | A1 FT | Only available from a correct split, not from an incomplete approach |
| Correct use of limits 0 and 6 in expression involving $p\ln(x^2+4)$, $q\tan^{-1}\left(\dfrac{x}{2}\right)$ and no incorrect terms | M1 | $p$ and $q$ should be constants. The $x$ term is not required at this stage |
| Obtain $6 + \dfrac{1}{2}\ln 10 - 2\tan^{-1}3$ | A1 | ISW. Or three term equivalent (must combine ln terms). Accept with $\dfrac{1}{2}\ln|10|$ |

**Alternative method for Question 5:**

| Answer | Mark | Guidance |
|--------|------|----------|
| Use substitution $x = 2\tan\theta$ to obtain $\int 2\tan^2\theta + \tan\theta \, d\theta$ | B1 | |
| Attempt integration and obtain $p\tan\theta$ or $r\ln(\sec\theta)$ from correct working | M1 | |
| Obtain $2\tan\theta(-2\theta)$ | A1 FT | Follow through on their coefficients after substitution |
| Obtain $\ln\sec\theta$ | A1 FT | Follow through on their coefficients after substitution |
| Use correct limits 0 and $\tan^{-1}3$ in expression involving $u\tan\theta$, $v\ln\sec\theta$ and no incorrect terms | M1 | $u$ and $v$ should be constants. The $\theta$ term is not required at this stage |
| Obtain $6 + \ln|\sec(\tan^{-1}3)| - 2\tan^{-1}3$ | A1 | ISW. Or three term equivalent. Not required to simplify $\ln|\sec(\tan^{-1}3)|$ |
| **Total** | **6** | |

---