Question

Evaluate the integral: integral of t sinh mt dt

Answer 1

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