Moderate -0.8 This is a straightforward integration by substitution question requiring only the standard technique of letting u = 2x - 1, rewriting the integral, and integrating a simple power. It's more routine than average A-level questions since it's a single-step application of a standard method with no complications or multi-part reasoning.
o.e., but must have \(+ c\) and single fraction mark final ans
Total: [4]
e.g. correct power of \((2x-1)\); e.g. \(\frac{3}{4}(2x-1)^{4/3}\) seen; so \(\frac{3}{8}(2x-1)^{4/3}\) is M1M1M1A0
let $u = 2x - 1$, $du = 2 \, dx$ | M1 | substituting $u = 2x - 1$ in integral
$\int \sqrt{2x-1} \, dx = \int \frac{1}{2}u^{1/3} \, du$ | M1 | $\times \frac{1}{2}$ o.e.
$= \frac{3}{8}u^{4/3} + c$ | M1 | integral of $u^{1/3} = u^{4/3}/(4/3)$ (oe) soi
$= \frac{3}{8}(2x-1)^{4/3} + c$ | A1 cao | o.e., but must have $+ c$ and single fraction mark final answer
**Total: [4]** | | so $\frac{3}{4}(2x-1)^{4/3} + c$ is M1M0M1A0
**or**
$\int \sqrt{2x-1} \, dx = \frac{1}{2} \times (2x-1)^{4/3} \div 4/3$ | M1 | $(2x-1)^{4/3}$ seen
$\frac{1}{2} \times (2x-1)^{4/3} \times 3/4$ | M1 | $\div 4/3$ (oe) soi
$= \frac{3}{8}(2x-1)^{4/3} + c$ | A1 cao | o.e., but must have $+ c$ and single fraction mark final ans
**Total: [4]** | | e.g. correct power of $(2x-1)$; e.g. $\frac{3}{4}(2x-1)^{4/3}$ seen; so $\frac{3}{8}(2x-1)^{4/3}$ is M1M1M1A0
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