Challenging +1.2 Parts (a) and (b) are routine proofs using standard hyperbolic identities requiring only algebraic manipulation and basic differentiation. Part (c) is a guided substitution integral that, while requiring careful execution, follows a clear path once the substitution is made. Part (d) is straightforward evaluation of definite integrals. This is a standard Further Maths question with multiple scaffolded parts, slightly above average difficulty due to the technical manipulation required but not requiring novel insight.
Using exponentials, show that \(\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1\).
By differentiating both sides of the identity in part (a) with respect to \(u\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\).
Use the substitution \(x = \sinh ^ { 2 } u\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form \(a \sinh ^ { - 1 } b \sqrt { x } + \mathrm { f } ( x )\) where \(a\) and \(b\) are integers and \(\mathrm { f } ( x )\) is a function to be determined.
Hence determine the exact area of the region between the curve \(y = \sqrt { \frac { x } { x + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(p + q \ln r\) where \(p , q\) and \(r\) are numbers to be determined.
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\begin{enumerate}[label=(\alph*)]
\item Using exponentials, show that $\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1$.
\item By differentiating both sides of the identity in part (a) with respect to $u$, show that $\sinh 2 u \equiv 2 \sinh u \cosh u$.
\item Use the substitution $x = \sinh ^ { 2 } u$ to find $\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x$. Give your answer in the form $a \sinh ^ { - 1 } b \sqrt { x } + \mathrm { f } ( x )$ where $a$ and $b$ are integers and $\mathrm { f } ( x )$ is a function to be determined.
\item Hence determine the exact area of the region between the curve $y = \sqrt { \frac { x } { x + 1 } }$, the $x$-axis, the line $x = 1$ and the line $x = 2$. Give your answer in the form $p + q \ln r$ where $p , q$ and $r$ are numbers to be determined.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q3 [10]}}