Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 16

To help you solve this, we consider the the integration by parts:
int u * dv = uv - int v* du
Let u = t and dv = sinh(mt) dt.
based from int t*sinh(mt) dt for int u*dv
In this integral, the "m" will be treated as constant since it is integrated with respect to "t".
From u = t , then du = dt
From dv = sinh(mt) dt , then int dv = v
In int sinh(mt) dt , let w = mt then dw= m dt or dt= (dw)/m
Substitute w = mt and dt = (dw)/m
int sinh(mt) dt = int (sinh(w)dw)/m
= (1/m) int sinh(w) dw
based from c is constant inint c f(x) dx=c int f(x) dx +C
(1/w) int sinh(w) dw = (1/w) cosh(w) +C
Substitute w = mt , it becomes v = 1/(m)cosh(mt)+C

Then:
u = t
du = dt
dv = sinh(mt) dt
v = 1/(m)cosh(mt)
Plug into the integration by parts: int u * dv = uv - int v* du
int t* sinh(mt) dt = t*1/(m)cosh(mt) - int 1/mcosh(mt) dt
= t/mcosh(mt) - 1/mint cosh(mt) dt
= t/mcosh(mt) - 1/m*1/msinh(mt)+C
= t/mcosh(mt) - 1/m^2 sinh(mt) +C
= (mtcosh(mt) -sinh(mt))/m^2 +C