I don't understand this thread. According to the OP
Addez123 said:
Homework Statement:: ##I = { \large \int \frac y {x^2+y^2} } dx~\dots~##
Since
##{~~~~}x = \left \{ \begin{align} & \text {variable of integration } \nonumber \\ &~~ \text {and is therefore the} \nonumber \\ & ~\text {independent variable} \nonumber \end{align} \right \}~~\Rightarrow~## ##\left \{ \begin{align} & y = y(x)~\text {because}~y~\text {is} \nonumber \\ & \text {normally used as the} \nonumber \\ & ~ \text {dependent variable} \nonumber \end{align} \right \}##
then
##{~~~~}I = { \large \int \frac y {x^2+y^2} } dx~\Rightarrow~I = { \large \int \frac {y(x)} {x^2+y^2} } dx##
so that integration by parts, wherein the right-hand side is a function of the independent variable only, no longer applies. But according to the forum mentor in post #2,
ergospherical said:
##\dots##. It would be correct to say, for example,\begin{align*}
\int \dfrac{1}{\frac{x^2}{y^2} + 1} dx = y \int \dfrac{1}{\frac{x^2}{y^2} + 1} d\left( \frac{x}{y} \right) = y\mathrm{arctan}\left( \frac{x}{y} \right)
\end{align*}
indicating that ##~y~## is actually a constant since
##{~~~~}d \left( { \large \frac {x}{y} } \right) = { \large \frac {dx}{y} }~\Rightarrow~y { \large \int } \dfrac{1}{\frac{x^2}{y^2} + 1} d\left( { \large \frac{x}{y} } \right) = { \large \int } \dfrac{1}{\frac{x^2}{y^2} + 1} dx##
If that is the case, then the original integral should have been written as
##{~~~~}I = { \large \int \frac k {x^2+k^2} } dx~\Leftarrow~k = \rm {constant}##
right at the start, or early on a remark posted by people who know more, in order to avoid confusion and waste of time.