Integrate sin(2sinh(3x)): Solutions & Tips

anthonybommarito1
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Homework Statement



∫sin(2sinh(3x))

Homework Equations

The Attempt at a Solution


okay so i did a u substitution letting u=3x so we get 1/3∫sin(2sinh(u)) but i have no idea how to get rid of the sinh, i tried writing in exponential form or maybe i have to use some identity.. I am not sure where to go from there!
 
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anthonybommarito1 said:

Homework Statement



∫sin(2sinh(3x))

Homework Equations

The Attempt at a Solution


okay so i did a u substitution letting u=3x so we get 1/3∫sin(2sinh(u)) but i have no idea how to get rid of the sinh, i tried writing in exponential form or maybe i have to use some identity.. I am not sure where to go from there!
You should always include the dx or whatever when writing an integral.

It's not clear this integral even has an anti-derivative which can be expressed in terms of elementary functions.

Where does this integral come from?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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