C4 January 2006 Q3
3. Using the substitution \(u^2 = 2x - 1\), or otherwise, find the exact value of
\[\int_1^5 \frac{3x}{\surd(2x - 1)}\,\mathrm{d}x.\]
(8)
| Scheme | Marks |
|---|---|
| Uses substitution to obtain \(x = \mathrm{f}(u)\ \left[\dfrac{u^2 + 1}{2}\right]\), | M1 |
| and to obtain \(u\dfrac{\mathrm{d}u}{\mathrm{d}x} = \) const. or equiv. | M1 |
| Reaches \(\displaystyle\int \frac{3(u^2 + 1)}{2u}u\,\mathrm{d}u\) or equivalent | A1 |
| Simplifies integrand to \(\displaystyle\int \left(3u^2 + \frac{3}{2}\right)\mathrm{d}u\) or equiv. | M1 |
| Integrates to \(\frac{1}{2}u^3 + \frac{3}{2}u\) | M1 A1ft |
| Uses new limits 3 and 1 substituting and subtracting (or returning to function of \(x\) with old limits) | M1 |
| To give 16 cso | A1 |
| (8) | |
| (8 marks) |
A1ft dependent on all previous Ms
“By Parts”
| Attempt at “right direction” by parts \(\left[3x(2x - 1)^{\frac{1}{2}}\right] - \left\{\displaystyle\int 3(2x - 1)^{\frac{1}{2}}\,\mathrm{d}x\right\}\) | M1 M1{M1A1} |
| \(\ldots\ldots\ldots\ldots - (2x - 1)^{\frac{3}{2}}\) | M1A1ft |
| Uses limits 5 and 1 correctly; \([42 - 26]\) 16 | M1A1 |