Standard +0.8 This is a challenging integration by substitution problem requiring multiple steps: applying the given substitution x = u² - 1, simplifying the resulting integral (likely involving partial fractions or logarithmic forms), and then substituting back to reach the specific form with nested radicals. The algebraic manipulation to achieve the exact form given, particularly with the nested square roots and the ratio structure, is non-trivial and goes beyond routine substitution exercises. However, it's guided by the suggested substitution and is a standard Pre-U/Further Maths technique rather than requiring novel insight.
11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that
$$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$
where \(A\) is an arbitrary constant and \(x > 0\).
Show correct use of at least one log law on a correct equation [M1]
State or show clearly \(+ c = \ln A\) and obtain \(\ln\left(A\sqrt{\dfrac{\sqrt{x+1}-1}{\sqrt{x+1}+1}}\right)\) AG [A1]
Total: [10]
Obtain $\mathrm{d}x = f(u)\,\mathrm{d}u$ or equivalent [B1]
Rewrite $\sqrt{x+1}$ in terms of $u$ and substitute to obtain an integral in $u$ [M1]
Obtain unsimplified $\displaystyle\int \frac{2u\,\mathrm{d}u}{2(u^2-1)\sqrt{(u^2-1)+1}}$ [A1]
Obtain $\displaystyle\int \frac{\mathrm{d}u}{u^2-1}$ [A1]
Use partial fractions in form $\dfrac{A}{u+1} + \dfrac{B}{u-1}$ [M1]
Obtain $A = -\dfrac{1}{2}$ and $B = \dfrac{1}{2}$ both correctly placed [A1]
Integrate to obtain $\left(k\ln|u-1| + m\ln|u+1|\right)$ [M1]
Obtain $\dfrac{1}{2}\ln(u-1) - \dfrac{1}{2}\ln(u+1) = c$ or $\dfrac{1}{2}\ln\left(\dfrac{u-1}{u+1}\right) + c$ [A1]
Show correct use of at least one log law on a correct equation [M1]
State or show clearly $+ c = \ln A$ **and** obtain $\ln\left(A\sqrt{\dfrac{\sqrt{x+1}-1}{\sqrt{x+1}+1}}\right)$ **AG** [A1]
**Total: [10]**