Finding the Expression for v Using Partial Integration

[LeitHunt]

Homework Statement

For an analytic function, f(x+iy)=u(x,y)+iv(x,y),u is given by u=(3x^2) -(3y^2). The expression for v, considering K to be a constant is?

Homework Equations

δu/δx=δv/δy
δu/δy=-δv/δx
[/B]

The Attempt at a Solution

My attempt :-
u=(3x^2) -(3y^2)

δu/δx=6x & δu/δy=-6y...(1)

From Relevant equations,

δu/δx=δv/δy

From (1)
6x=δv/δy

∫δv=∫6x δy

v=6xy+f(x)

From Relevant equations,

δu/δy=-δv/δx
From (1)
-6y=-δv/δx

-6y=-[δ/δx(6xy+f(x)]

6y=6y+f'(x)

f'(x)=0

Integrating,

f(x)=0+K

∴v=6xy+0+K

v=6xy+K[/B]

https://m.imgur.com/a/CDQlL
I know Partial derivative but never heard about Partial integration so first time came across this type of problem.
I checked the procedure on Internet and try to solve according to that.
In book they may have solved it wrong.
In picture on right side of red line I solved the problem according to Internet procedures and I got the same answer as book. But in book the (6x) should be partially integrate with respect to y by mistake they may have integrated with respect to x.
Which one is correct in picture solved by book or by me?

 

[Ray Vickson]

Homework Statement

For an analytic function, f(x+iy)=u(x,y)+iv(x,y),u is given by u=(3x^2) -(3y^2). The expression for v, considering K to be aconstant is?

Homework Equations

δu/δx=δv/δy
δu/δy=-δv/δx
[/B]

The Attempt at a Solution

https://imgur.com/a/S82cs[/B]
I know Partial derivative but never heard about Partial integration so first time came across this type of problem.
I checked the procedure on Internet and try to solve according to that.
In book they may have solved it wrong.
In picture on right side of red line I solved the problem according to Internet procedures and I got the same answer as book. But in book the (6x) should be partially integrate with respect to y by mistake they may have integrated with respect to x.
Which one is correct in picture solved by book or by me?

Do not post images---especially sideways ones. Just type out the solution (which is the PF standard!)

You should not expect people to lie down sideways on their desks in order to read your pictures.

 

[LeitHunt]

Do not post images---especially sideways ones. Just type out the solution (which is the PF standard!)

You should not expect people to lie down sideways on their desks in order to read your pictures.

Sorry I was on mobile. I'll edit it soon :)

 

[LeitHunt]

Edited :)

 

[Dick]

Edited :)

You are correct. Clearly, the book integrated part i) wrong.