Inverse hyperbolic functionsEdexcel A-Level Further Maths: Flashcards

What these 13 flashcards ask

Domain and range of \operatorname{arsinh}x?
Domain and range of \operatorname{arcosh}x?
Domain and range of \operatorname{artanh}x?
Logarithmic form of \operatorname{arsinh}x?
Logarithmic form of \operatorname{arcosh}x?
Logarithmic form of \operatorname{artanh}x?
How do you start deriving \operatorname{arsinh}x?
Why is the root x-\sqrt{x^2+1} rejected in the arsinh derivation?
Why take the plus sign in the arcosh derivation?
Exact value of \operatorname{arsinh}\frac34?
\int\frac1{\sqrt{x^2+a^2}}\,dx?
\int\frac1{\sqrt{x^2-a^2}}\,dx?
Which substitution suits \int\frac1{\sqrt{x^2-a^2}}\,dx?

Exam questions on Inverse hyperbolic functions

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
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).