2

I'm reviewing multivariable integrals, and the constants are confusing me. If I have:

$$ f(x, y) = \int \frac{\partial f(x,y)}{\partial x} dx $$ $$ f(x, y) = \int 2xy \,dx $$ factor out $y$ which is treated as a constant. $$ f(x, y) = y \int 2x \,dx $$

But, I expected

$$ f(x, y) = 2y \int x \,dx $$

Since $2$ is also a constant. The question is, why factor out $y$ and not $2$ as well. What am I missing?

Michael Hardy
  • 1
maximus
  • 125
  • 2
    you are right, you can factor 2 out as well. Even if you don't, the last two integrals that you've written are same. –  Aug 11 '14 at 22:11

1 Answers1

2

It is perfectly correct to pull out the $2$. But $$ \int 2x\,dx= x^2+\text{constant} $$ and $$ 2\int x\,dx = 2\left( \frac {x^2}{2}\right)+\text{constant} = x^2+\text{constant} $$ so the first way is a bit simpler.

Michael Hardy
  • 1