## Question 9:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $u = x^2 + 1$, $dv = e^{2x}$ or $u = x^2$, $dv = e^{2x}$ | M1 | $1^{\text{st}}$ stage $= f(x) +/- \int g(x)\, dx$ |
| $1^{\text{st}}$ stage $= \frac{1}{2}(x^2+1)e^{2x} - \int x\,e^{2x}\,dx$ or $\frac{1}{2}x^2e^{2x} - \int xe^{2x}\,dx$ | A1 | |
| For $\int x\,e^{2x}\,dx$, use $u = x$, $dv = e^{2x}$ | M1 | ditto |
| $= \frac{1}{2}x\,e^{2x} - \frac{1}{4}e^{2x}$ | A1 | Tolerate second sign error in $-\int xe^{2x}\,dx$ |
| Complete final stage $= \frac{1}{2}(x^2+1)e^{2x} - \frac{1}{4}(2x-1)e^{2x}$ | A1 | soi; may be separate terms |
| Correct (method) use of limits seen anywhere | M1 | Do not accept $(\ldots) - 0$ |
| Final answer $= \frac{3}{4}e^2 - \frac{3}{4}$ | A1 | ISW; if A0, answer of $4.79\ldots \rightarrow$ M1 |
| **[7]** | | |

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