Standard +0.8 This is a Further Maths question requiring knowledge of inverse hyperbolic/trig integrals (specifically arcsinh form), careful algebraic manipulation to extract constants, evaluation at limits, and combining logarithms to find an exact answer. While the technique is standard for FP2, the multi-step calculation with two integrals and exact logarithmic form elevates it slightly above average difficulty.
For either integral; allow attempt at ln version here
Get \(\frac{1}{3}\sinh^{-1}\frac{3}{4}x\)
A1
Or ln version
Get \(\frac{1}{2}\sinh^{-1}\frac{1}{2}x\)
A1
Or ln version
Use limits in their answers
M1
Attempt to use correct ln laws to set up a solvable equation in \(a\)
M1
Get \(a = 2^{1/2} \cdot 3^{1/2}\)
A1
Or equivalent
## Question 6:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Get $k\sinh^{-1}k_1 x$ | M1 | For either integral; allow attempt at ln version here |
| Get $\frac{1}{3}\sinh^{-1}\frac{3}{4}x$ | A1 | Or ln version |
| Get $\frac{1}{2}\sinh^{-1}\frac{1}{2}x$ | A1 | Or ln version |
| Use limits in their answers | M1 | |
| Attempt to use correct ln laws to set up a solvable equation in $a$ | M1 | |
| Get $a = 2^{1/2} \cdot 3^{1/2}$ | A1 | Or equivalent |
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