How would one integrate this: $\int xdy$, where y is a function of x?

Question

How would one go about integrating $\int xdy$ if y is a function of x? A teacher told me this equals $xy - \int ydx$ - is this true, and if so, what is the proof for it? Thanks!

Answer 1

Hint...write $dy=\frac{dy}{dx}dx$ and then do integration by parts

Answer 2

$\int f(x)dy$ where $y$ is a function of $x$ can be interpreted as a Riemann-Stiltjes integral, basically, provided $y$ is increasing, you can define a Riemann-Stiltjies sum as

$$ S_N=\sum_{k=0}^Nf(x_k)(y(x_{k+1})-y(x_k))$$

where the $x_k$ is some subdivision of the interval you're considering for the integration that gets finer as $N$ increases. Note that if you take $y(x)=x$ you just obtain the regular Riemann sum. You can show that under certain conditions this sum converges and we note

$$ \int f(x)dy=\lim_{N\to\infty}S_N$$

Now it's easy to see that if $y$ is differentiable then you have

$$ \int f(x)dy=\int f(x)y'(x)dx$$

to see it just divide and multiply by $x_{k+1}-x_k$ in the sum

By integration by parts it is now easy to see that if $y$ is differentiable and increasing

$$ \int xdy=\int xy'dx=xy-\int ydx$$

If you're not familiar with integration by parts, just notice that $$ (fg)'=f'g+fg'$$ taking $f=x$ and $g=y$, and integrating both sides, can you arrive at the equality?