FP3 June 2010 Q4
4. \[I_n = \int_0^{a} (a - x)^n\cos x\,\mathrm{d}x, \qquad a > 0, \quad n \geqslant 0\]
(a) Show that, for \(n \geqslant 2\), \[I_n = na^{n-1} - n(n - 1)I_{n-2}\] (5)
(b) Hence evaluate \(\displaystyle\int_0^{\frac{\pi}{2}} \left(\frac{\pi}{2} - x\right)^2\cos x\,\mathrm{d}x\). (3)
| Scheme | Marks |
|---|---|
| \(\displaystyle\int \left(a - x\right)^n\cos x\,\mathrm{d}x = \left(a - x\right)^n\sin x + \displaystyle\int n\left(a - x\right)^{n-1}\sin x\,\mathrm{d}x\) | M1A1 |
| \(\left[\left(a - x\right)^n\sin x\right]_0^{a} = 0\) | A1 |
| \(= -n\left(a - x\right)^{n-1}\cos x - \displaystyle\int n\left(n - 1\right)\left(a - x\right)^{n-2}\cos x\,\mathrm{d}x\) | dM1 |
| \(\mathrm{I}_n = na^{n-1} - n(n - 1)\mathrm{I}_{n-2}\) * | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(\mathrm{I}_2 = 2\left(\dfrac{\pi}{2}\right) - 2\displaystyle\int_0^{\frac{\pi}{2}} \cos x\,\mathrm{d}x\) | M1 A1 |
| \(= \pi - 2\left[\sin x\right]_0^{\frac{\pi}{2}} = \pi - 2\) | A1 |
| (3) | |
| (8 marks) |