The integral `int (1/sqrt 2)V_0*e^(-kt)*sqrt(1+e^(-kt)) dt` has to be determined.

Let `1+e^(-kt) = y`

`dy/dt = -k*e^(-kt)`

=> `e^(-kt) dt = (-1/k)dy`

`int (1/sqrt 2)V_0*e^(-kt)*sqrt(1+e^(-kt)) dt`

=> `int (1/sqrt 2)V_0*(-1/k)sqrt(y) dy`

=> `-V_0/(sqrt 2*k)int sqrt(y) dy`

=> `-V_0/(sqrt 2*k)y^(3/2)/(3/2)`

=> `(-2*V_0)/(3*sqrt 2*k)y^(3/2)`

Substitute `y = 1+e^(-kt)`

=> `(-2*V_0)/(3*sqrt 2*k)(1+e^(-kt))^(3/2)`

**The required integral `int (1/sqrt 2)V_0*e^(-kt)*sqrt(1+e^(-kt)) dt = (-2*V_0)/(3*sqrt 2*k)(1+e^(-kt))^(3/2)` **

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