Standard +0.3 This is a straightforward substitution question with clear guidance (u given explicitly). Students must find du/dx, change limits, express x² in terms of u, and integrate a polynomial. While it requires careful algebraic manipulation and is worth several marks, it follows a standard template with no conceptual surprises, making it slightly easier than average.
Attempt to connect \(dx\) & \(du\) or find \(\frac{dx}{du}\) or \(\frac{du}{dx}\)
M1
No accuracy; not 'by parts'
\(dx = 2u\, du\) or \(\frac{du}{dx} = \frac{1}{2}(x+2)^{-\frac{1}{2}}\)
A1
AEF
Indefinite integral \(\rightarrow \int 2(u^2-2)^2 \left(\frac{u}{u}\right)(du)\)
A1
May be implied later
Cancel \(u/u\) and attempt to square out
M1
\(\int kI\,(du)\) where \(k=2\) or \(\frac{1}{2}\) or 1 and \(I=(u^2-2)^2\) or \((2-u^2)^2\) or \((u^2+2)^2\)
dep on previous M1
Attempt to change limits if working with \(f(u)\) after integration
M1
or re-substitute into integral and use \(-1\) & \(7\)
Indefinite integral \(= \frac{2}{5}u^5 +/- \frac{8}{3}u^3 + 8u\) or \(\frac{1}{10}u^5 +/- \frac{2}{3}u^3 + 2u\)
A1
or \(\frac{1}{5}u^5 +/- \frac{4}{3}u^3 + 4u\)
\(\frac{652}{15}\) or \(43\frac{7}{15}\)
A1
ISW but no '+c'
Total: 7 marks
# Question 4:
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempt to connect $dx$ & $du$ or find $\frac{dx}{du}$ or $\frac{du}{dx}$ | M1 | No accuracy; not 'by parts' |
| $dx = 2u\, du$ or $\frac{du}{dx} = \frac{1}{2}(x+2)^{-\frac{1}{2}}$ | A1 | AEF |
| Indefinite integral $\rightarrow \int 2(u^2-2)^2 \left(\frac{u}{u}\right)(du)$ | A1 | May be implied later |
| Cancel $u/u$ and attempt to square out | M1 | |
| $\int kI\,(du)$ where $k=2$ or $\frac{1}{2}$ or 1 and $I=(u^2-2)^2$ or $(2-u^2)^2$ or $(u^2+2)^2$ | | dep on previous M1 |
| Attempt to change limits if working with $f(u)$ after integration | M1 | or re-substitute into integral and use $-1$ & $7$ |
| Indefinite integral $= \frac{2}{5}u^5 +/- \frac{8}{3}u^3 + 8u$ or $\frac{1}{10}u^5 +/- \frac{2}{3}u^3 + 2u$ | A1 | or $\frac{1}{5}u^5 +/- \frac{4}{3}u^3 + 4u$ |
| $\frac{652}{15}$ or $43\frac{7}{15}$ | A1 | ISW but no '+c' |
**Total: 7 marks**
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