FP3 June 2011 Q3
3. Show that
(a) \(\displaystyle\int_5^8 \frac{1}{x^2 - 10x + 34}\,\mathrm{d}x = k\pi\), giving the value of the fraction \(k\), (5)
(b) \(\displaystyle\int_5^8 \frac{1}{\sqrt{(x^2 - 10x + 34)}}\,\mathrm{d}x = \ln(A + \sqrt{n})\), giving the values of the integers \(A\) and \(n\). (4)
| Scheme | Marks |
|---|---|
| \(x^2 - 10x + 34 = (x - 5)^2 + 9\) so \(\dfrac{1}{x^2 - 10x + 34} = \dfrac{1}{(x - 5)^2 + 9} = \dfrac{1}{u^2 + 9}\) (mark can be earned in either part (a) or (b)) | B1 |
| \(I = \displaystyle\int \dfrac{1}{u^2 + 9}\,\mathrm{d}u = \left[\dfrac{1}{3}\arctan\left(\dfrac{u}{3}\right)\right]\) or \(I = \displaystyle\int \dfrac{1}{(x - 5)^2 + 9}\,\mathrm{d}u = \left[\dfrac{1}{3}\arctan\left(\dfrac{x - 5}{3}\right)\right]\) | M1 A1 |
| Uses limits 3 and 0 to give \(\dfrac{\pi}{12}\) or Uses limits 8 and 5 to give \(\dfrac{\pi}{12}\) | DM1 A1 |
| (5) |
Notes
B1 CAO allow recovery in (b)
1M1 Integrating getting k arctan term
1A1 CAO
2DM1 Correctly using limits.
2A1 CAO
| Scheme | Marks |
|---|---|
| Alt 1 \(I = \ln\left(\left(\dfrac{x - 5}{3}\right) + \sqrt{\left(\dfrac{x - 5}{3}\right)^2 + 1}\right)\) or \(I = \ln\left(\dfrac{x - 5 + \sqrt{(x - 5)^2 + 9}}{3}\right)\) or \(I = \ln\left((x - 5) + \sqrt{(x - 5)^2 + 9}\right)\) | M1 A1 |
| Uses limits 5 and 8 to give \(\ln(1 + \sqrt{2})\). | DM1 A1 |
| (4) | |
| (9 marks) |
Notes
(b) Alt 2
| Scheme | Marks |
|---|---|
| Uses \(u = x - 5\) to get \(I = \displaystyle\int \dfrac{1}{\sqrt{u^2 + 9}}\,\mathrm{d}u = \left[\mathrm{arsinh}\left(\dfrac{u}{3}\right)\right] = \ln\left\{u + \sqrt{u^2 + 9}\right\}\) | M1 A1 |
| Uses limits 3 and 0 and ln expression to give \(\ln(1 + \sqrt{2})\). | DM1 A1 |
| (4) |
(b) Alt 3
| Scheme | Marks |
|---|---|
| Use substitution \(x - 5 = 3\tan\theta,\ \dfrac{\mathrm{d}x}{\mathrm{d}\theta} = 3\sec^2\theta\) and so \(I = \displaystyle\int \sec\theta\,\mathrm{d}\theta = \ln(\sec\theta + \tan\theta)\) | M1 A1 |
| Uses limits 0 and \(\dfrac{\pi}{4}\) to get \(\ln(1 + \sqrt{2})\). | DM1 A1 |
| (4) |
1M1 Integrating to get a ln or hyperbolic term
1A1 CAO
2DM1 Correctly using limits.
2A1 CAO