# int t sinh(mt) dt Evaluate the integral

## Expert Answers 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

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