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...
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 in`int 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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