Can somebody please help me with this math problem with intergrals? 
I really don´t know how to solve this and I´ve been sitting with this for days now. 

I´ve tried to solve it with partly intergration where I choose the first part to be f(x) and the other part to be g´(x) and tried to use the product rule to solve them, but it´s becoming even more complex as I go by...

I would be so happy if I can get some help or tips on how to solve this.

Image (1 of 1)

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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)`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial