I am stuck on taking the integral of this:
$$\int \sin(9\sinh(2x))\,{\rm d}x$$
I tried using using a double $u$ substitution, but i got confused. Nothing seems to cancel out.
Problem with integral of hyperbolic function
-2
$\begingroup$
integration
hyperbolic-functions
-
1This does not look like an integral that has an elementrary antiderivative. – 2017-02-04
-
0how would i start solving it? – 2017-02-04
-
0You can likely find a solution in the form of a power-series. I might be wrong, but I won't expect this to have a simple closed form solution. – 2017-02-04
-
0See [Bessel and Struve functions](http://math.stackexchange.com/questions/1196401). – 2017-02-04
1 Answers
0
In general, $~\displaystyle\int_0^\infty\sin(a~\sinh t)~dt~=~\dfrac\pi2~\Big(I_0(a)-L_0(a)\Big),~$ where I and L represent the Bessel and Struve functions, respectively. Letting $t=2x$ and $a=9$ should therefore yield a closed form expression for the definite integral in terms of these two special functions. However, evaluating the indefinite integral would require the existence of “incomplete” Bessel and Struve functions, which, unfortunately, do not formally exist $($as a conventional object of study within the larger realm of mathematics$)$.