\sinh^2x in terms of \cosh2x?

\sinh^2x=\frac12\left(\cosh2x-1\right)

\cosh^2x in terms of \cosh2x?

\cosh^2x=\frac12\left(\cosh2x+1\right)

How do you integrate an inverse function?

Integrate by parts with u=f(x) and \frac{dv}{dx}=1.

\int\operatorname{arsinh}x\,dx?

x\operatorname{arsinh}x-\sqrt{1+x^2}+c

x\arctan x-\frac12\ln\left(1+x^2\right)+c

Integrating hyperbolic and inverse functionsEdexcel International A Level Further Maths: Flashcards

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$\int\sinh x\,dx$?

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$\int\sinh x\,dx$ ?: $\cosh x+c$
$\int\cosh x\,dx$ ?: $\sinh x+c$
$\int\operatorname{sech}^2x\,dx$ ?: $\tanh x+c$
$\int\tanh x\,dx$ ?: $\ln\cosh x+c$
$\int\cosh kx\,dx$ ?: $\frac1k\sinh kx+c$
$\sinh^2x$ in terms of $\cosh2x$ ?: $\sinh^2x=\frac12\left(\cosh2x-1\right)$
$\cosh^2x$ in terms of $\cosh2x$ ?: $\cosh^2x=\frac12\left(\cosh2x+1\right)$
$\int x\cosh x\,dx$ ?: $x\sinh x-\cosh x+c$
$\int x\sinh x\,dx$ ?: $x\cosh x-\sinh x+c$
How do you integrate an inverse function?: Integrate by parts with $u=f(x)$ and $\frac{dv}{dx}=1$ .
$\int\operatorname{arsinh}x\,dx$ ?: $x\operatorname{arsinh}x-\sqrt{1+x^2}+c$
$\int\arctan x\,dx$ ?: $x\arctan x-\frac12\ln\left(1+x^2\right)+c$
$\operatorname{arsinh}x$ in logarithmic form?: $\ln\left(x+\sqrt{x^2+1}\right)$

Exam questions on Integrating hyperbolic and inverse functions

A student is working with the integral $I=\int_0^{\ln2}\cosh2x\,dx$ .
Hence find the exact value of $\int_0^{\ln2}\sinh^2x\,dx$ .2 marks
Integration by parts is used to integrate the product of $x$ with a hyperbolic function.
Find $\int x\sinh x\,dx$ .2 marks
Let $I=\int_0^{4/3}\operatorname{arsinh}x\,dx$ .
Use integration by parts to find $\int\operatorname{arsinh}x\,dx$ .3 marks

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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).