Can A Dependent Variable be Factored Out of Integral?

Question

say I have a the equation $$y = xw + z$$

And I am trying to compute $$\begin{aligned}\int \text{exp}((y - xw)^2 - w^2)dw &= \int\text{exp}(y^2 - 2xwy + x^2w^2 - w^2)dw \\&= \int\text{exp}(y^2)exp(-2xwy + x^2 -w^2)dw \end{aligned}$$

Am I allowed to factor out the $\text{exp}(y^2)$ term considering $y = xw + z$ to get $$\text{exp}(y^2)\int\text{exp}(-2xwy + x^2 - w^2)dw$$

I appreciate all of the swift replies. I did believe that was not correct, but needed a sanity check. Thank you all. To answer some comments, x and z are constants here.

No, for the same reason that if $y=x^2$ you cannot say that $\int ydx=y\int dx=yx+C$. — morrowmh, Jul 28 '22 at 22:51
You have $y-xw=z$ and therefore the integral is $\int\exp(z^2-w^2)dw=\exp(z^2)\int\exp(-w^2)dw.$ — David K, Jul 29 '22 at 13:05

score 1 · Answer 1 · answered Jul 28 '22 at 22:58

1

No, you can't factor out $\exp{\left(y^2\right)}$ because $y$ depends on $w$. If it was independent, like say if $y$ is a constant or if $y$ equals anything that doesn't involve $w$, then yes you can pull it out.

answered Jul 28 '22 at 22:58

Accelerator

4,923

score 0 · Answer 2 · edited Jul 29 '22 at 06:12

0

Sorry, not enough reputation to post this as a comment : my answer is that you cannot factor out $\exp(y^2)$ because it's dependent on $w$. But can you give more details about your variables ? are there any constant ?

edited Jul 29 '22 at 06:12

MafPrivate

4,033

answered Jul 28 '22 at 22:53

mimi

11

Can A Dependent Variable be Factored Out of Integral?

2 Answers2