Inverse hyperbolic functionsEdexcel A-Level Further Maths: Mind map

Exam questions on Inverse hyperbolic functions

1. The inverse hyperbolic functions have the logarithmic forms $\operatorname{arsinh}x=\ln\left(x+\sqrt{x^2+1}\right)$, $\operatorname{arcosh}x=\ln\left(x+\sqrt{x^2-1}\right)$ and $\operatorname{artanh}x=\frac12\ln\frac{1+x}{1-x}$.
   Find the exact value of $\operatorname{artanh}\frac13$, giving your answer as a multiple of $\ln2$.2 marks
2. Let $y=\operatorname{arcosh}x$ for $x\ge1$, so that $x=\cosh y$ with $y\ge0$.
   Find the exact value of $\operatorname{arcosh}\frac{13}{5}$.2 marks
3. Let $y=\operatorname{arsinh}x$, so that $x=\sinh y$, where $\sinh y=\frac{e^{y}-e^{-y}}{2}$.
   Derive the logarithmic form $\operatorname{arsinh}x=\ln\left(x+\sqrt{x^2+1}\right)$.3 marks