# Thread: What did I do wrong? (integration)

1. ## What did I do wrong? (integration)

I was playing around and did this:
$\displaystyle f(x)g(x)=f_xg_x$
Then:
$\displaystyle \frac{d}{dx}\left [f_xg_x\right ]=\frac{g_xdf_x}{dx}+\frac{f_xdg_x}{dx}$
so
$\displaystyle f_xg_x=\int \frac{g_xdf_x}{dx}+\frac{f_xdg_x}{dx}dx=\int \frac{g_xdf_x}{dx}dx+\int \frac{f_xdg_x}{dx}dx=\int g_xdf_x+\int f_xdg_x$
so
$\displaystyle f_xg_x=g_x \int df_x+f_x \int dg_x$
and
$\displaystyle f_xg_x=f_xg_x+f_xg_x=2f_xg_x$

Any obvious division by zero someone should point out?
Or do I just not understand the concept of something?

2. Are you trying to derive the integration by parts formula?

Let $\displaystyle u = u(x)$ and $\displaystyle v = v(x)$.

By the product rule:

$\displaystyle \frac{d}{dx}(uv) = u\,\frac{dv}{dx} + v\,\frac{du}{dx}$

$\displaystyle uv = \int{\left(u\,\frac{dv}{dx} + v\,\frac{du}{dx}\right)\,dx}$

$\displaystyle uv = \int{u\,\frac{dv}{dx}\,dx} + \int{v\,\frac{du}{dx}\,dx}$

$\displaystyle uv = \int{u\,dv} + \int{v\,du}$

$\displaystyle \int{u\,dv} = uv - \int{v\,du}$.

3. But, I thought since in: udv, you where taking the integral of u with respect of dv and thus you could use the scalar multiple rule.
Samne thing with:
$\displaystyle \int ydx=yx+C$

4. It's not taking the integral of $\displaystyle u$ with respect to $\displaystyle v$.

It's just shorthand for $\displaystyle \int{u\,\frac{dv}{dx}\,dx}$. Do you see how it "looks like" the $\displaystyle dx$s cancel?

So you're taking the integral of the product of $\displaystyle u$ and $\displaystyle \frac{dv}{dx}$, with respect to $\displaystyle x$.