`(dr)/(dt) = 10e^t/sqrt(1-e^(2t))` Solve the differential equation.

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`(dr)/dt=10e^t/sqrt(1-e^(2t))`

`r=int10e^t/sqrt(1-e^(2t))dt`

Take the constant out,

`r=10inte^t/sqrt(1-e^(2t))dt`

Apply integral substitution : `u=e^t`

`du=e^tdt`

`r=10int1/sqrt(1-u^2)du`

Use the common integral :`int1/sqrt(1-u^2)du=arcsin(u)`

`r=10arcsin(u)`

Substitute back `u=e^t` and add a constant C to the solution,

`r=10arcsin(e^t)+C`

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`(dr)/dt=10e^t/sqrt(1-e^(2t))`

`r=int10e^t/sqrt(1-e^(2t))dt`

Take the constant out,

`r=10inte^t/sqrt(1-e^(2t))dt`

Apply integral substitution : `u=e^t`

`du=e^tdt`

`r=10int1/sqrt(1-u^2)du`

Use the common integral :`int1/sqrt(1-u^2)du=arcsin(u)`

`r=10arcsin(u)`

Substitute back `u=e^t` and add a constant C to the solution,

`r=10arcsin(e^t)+C`

 

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