Saturday, November 21, 2015

`int t sinh(mt) dt` Evaluate the integral

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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