FP3 June 2012 Q5
5.
(a) Differentiate \(x\,\mathrm{arsinh}\,2x\) with respect to \(x\). (3)
(b) Hence, or otherwise, find the exact value of \[\int_0^{\sqrt{2}} \mathrm{arsinh}\,2x\,\mathrm{d}x\] giving your answer in the form \(A\ln B + C\), where \(A\), \(B\) and \(C\) are real. (7)
| Scheme | Marks |
|---|---|
| \(\mathrm{arsinh}\,2x, + x\dfrac{2}{\sqrt{1 + 4x^2}}\) | M1A1, A1 |
| (3) |
Notes
a1M1: Differentiating getting an arsinh term and a term of the form \(\dfrac{px}{\sqrt{1 \pm qx^2}}\)
a1A1: cao \(\mathrm{arsinh}\,2x\)
a2A1: cao \(+\dfrac{2x}{\sqrt{1 + 4x^2}}\)
| Scheme | Marks |
|---|---|
| \(\therefore \displaystyle\int_0^{\sqrt{2}} \mathrm{arsinh}\,2x\,\mathrm{d}x = \left[x\,\mathrm{arsinh}\,2x\right]_0^{\sqrt{2}} - \displaystyle\int_0^{\sqrt{2}} \dfrac{2x}{\sqrt{1 + 4x^2}}\,\mathrm{d}x\) | 1M1 1A1ft |
| \(= \left[x\,\mathrm{arsinh}\,2x\right]_0^{\sqrt{2}} - \left[\dfrac{1}{2}(1 + 4x^2)^{\frac{1}{2}}\right]_0^{\sqrt{2}}\) | 2M1 2A1 |
| \(= \sqrt{2}\,\mathrm{arsinh}\,2\sqrt{2} - \left[\tfrac{3}{2} - \tfrac{1}{2}\right]\) | 3DM1 |
| \(= \sqrt{2}\ln(3 + 2\sqrt{2}) - 1\) | 4M1 3A1 |
| (7) | |
| (10 marks) |
Notes
b1M1: rearranging their answer to (a). OR setting up parts
b1A1: ft from their (a) OR setting up parts correctly
b2M1: Integrating getting an arsinh or arcosh term and a \(\left(1 \pm ax^2\right)^{\frac{1}{2}}\) term o.e..
b2A1: cao
b3DM1: depends on previous M, correct use of \(\sqrt{2}\) and 0 as limits.
b4M1: converting to log form.
b3A1: cao depends on all previous M marks.