Evaluate the integral: integral of t sinh mt dt

Integrate by parts

f = t, df = dt,

dg = sinh (mt) dt

g = cosh (mt) / m

put them together

tcosh (mt) / m - / m * ∫cosh (mt) dt

u = mt, du = mdt

tcosh (mt) / m - / m^2 * ∫cosh (u) dt

integral of cosh (u) = sinh (u)

tcosh (mt) / m - sinh (u) / m^2 + c1

substitute back in the u (mt)

tcosh (mt) / m - sinh (mt) / m^2 + c1

You can simplify this

((mt*cosh (mt) - sinh (mt))) / m^2