Moderate -0.5 This is a straightforward integration by substitution question requiring the standard technique of letting u = 2x - 1, then integrating u^(1/3) and adjusting for the constant factor. It's slightly easier than average as it's a direct application of a single technique with no complications, though it does require careful handling of the fractional power and the factor of 2.
substituting \(u = 2x - 1\) in integral; i.e. \(u^{1/3}\) or \(\sqrt[3]{u}\) seen in integral
M1
\(\times \frac{1}{2}\) o.e.; condone no \(du\), or \(dx\) instead of \(du\)
M1
integral of \(u^{1/3} = \frac{u^{4/3}}{4/3}\) (o.e.) soi; not \(x^{1/3}\)
A1cao
o.e., but must have \(+ c\) and single fraction; mark final answer \(\frac{3}{8}(2x-1)^{4/3} + c\)
Method 2: Direct integration
Answer
Marks
M1
\((2x - 1)^{4/3}\) seen
M1
\(\frac{4}{3}\) (o.e.) soi; e.g. \(\frac{3}{4}(2x - 1)^{4/3}\) seen
M1
\(\times \frac{1}{2}\)
A1cao
o.e., but must have \(+ c\) and single fraction; mark final answer \(\frac{3}{8}(2x-1)^{4/3} + c\)
Note: so \(\frac{3}{8}(2x-1)^{4/3} + c\) is M1M0M1A0; and \(\frac{3}{8}(2x-1)^{4/3}\) is M1M1M1A0
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# Question 1
**Method 1: Substitution**
M1 | substituting $u = 2x - 1$ in integral; i.e. $u^{1/3}$ or $\sqrt[3]{u}$ seen in integral
M1 | $\times \frac{1}{2}$ o.e.; condone no $du$, or $dx$ instead of $du$
M1 | integral of $u^{1/3} = \frac{u^{4/3}}{4/3}$ (o.e.) soi; not $x^{1/3}$
A1cao | o.e., but must have $+ c$ and single fraction; mark final answer $\frac{3}{8}(2x-1)^{4/3} + c$
**Method 2: Direct integration**
M1 | $(2x - 1)^{4/3}$ seen
M1 | $\frac{4}{3}$ (o.e.) soi; e.g. $\frac{3}{4}(2x - 1)^{4/3}$ seen
M1 | $\times \frac{1}{2}$
A1cao | o.e., but must have $+ c$ and single fraction; mark final answer $\frac{3}{8}(2x-1)^{4/3} + c$
**Note:** so $\frac{3}{8}(2x-1)^{4/3} + c$ is M1M0M1A0; and $\frac{3}{8}(2x-1)^{4/3}$ is M1M1M1A0
[4]