## Question 4:

| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration and reach $ax\tan^{-1}\frac{1}{2}x + b\int x\cdot\frac{1}{c+x^2}\,dx$ | \*M1 | OE. Denominator might be $1+\frac{x^2}{4}$ or $2+\frac{x^2}{2}$ |
| Obtain $x\tan^{-1}\!\left(\frac{1}{2}x\right) - \int x\cdot\frac{2}{4+x^2}\,dx$ | A1 | OE |
| Complete integration and obtain $x\tan^{-1}\!\left(\frac{1}{2}x\right) - \ln(4+x^2)$ | A1 | OE e.g. with $\ln\!\left(1+\frac{x^2}{4}\right)$ |
| Substitute limits correctly in an expression of the form $px\tan^{-1}x + q\ln(c+x^2)$ | DM1 | $2\tan^{-1}1 - \ln 8 + \ln 4$ OE |
| Obtain final answer $\frac{1}{2}\pi - \ln 2$ | A1 | OE exact answer. Needs a value for $\tan^{-1}1$ and a single log term |
| **Alternative method:** Use substitution $\theta = \tan^{-1}\frac{x}{2}$ to obtain $\lambda\int 2\theta\sec^2\theta\,d\theta$ and reach $p\theta\tan\theta + q\int\tan\theta\,d\theta$ | \*M1 | |
| Obtain $2\theta\tan\theta - 2\int\tan\theta\,d\theta$ | A1 | OE |
| Complete integration and obtain $2\theta\tan\theta + 2\ln(\cos\theta)$ | A1 | OE |
| Substitute correct limits correctly in an expression of the form $r\theta\tan\theta + s\ln(\cos\theta)$ | DM1 | Limits should be $\frac{\pi}{4}$ and 0. Limits must be in radians. |
| Obtain final answer $\frac{1}{2}\pi - \ln 2$ | A1 | OE exact answer. Need values for trig. functions and a single log term. |

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