Simple integral, can't get the right answer....

[Addez123]

Homework Statement

$$\int \frac y {x^2+y^2} dx$$

Relevant Equations

Just math

$$\int \frac y {x^2+y^2} dx$$
$$\frac 1 y * \int \frac 1 {\frac {x^2}{y^2} + 1} dx = \frac 1 y * atan(x/y)$$

The answer is just atan(x/y), which you get using u-substitution but I honestly don't see why I don't get it doing it the normal way.

 

[ergospherical]

It's because $\displaystyle{\int} \dfrac{1}{\frac{x^2}{y^2} + 1} dx \neq \mathrm{arctan}\left( \dfrac{x}{y} \right)$. 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*}

 

[PeroK]

It was already in standard integral format. Try differentiating your answer and you'll see that's not right.

 

[kumusta]

I don't understand this thread. According to the OP

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,

$\dots$. 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.