Rules of integrating by parts? In my textbook the rule for integrating by parts is 
but on a website it is stated that the rule for integrating by parts is 

Just wanted to confirm that they are the same rule, right?
Could someone show me how they are the same equation?
Many thanks!
 A: If you change $v$ to $\frac {dv} {dx}$ in the second  you get the first.
A: Both are same because your first image take $\frac{dv}{dx}$ in the place of $v$ (which is in the second). 
$\int u \frac{dv}{dx} dx = u \int \frac{dv}{dx} dx - \int u'(\int \frac{dv}{dx} dx) dx= uv - \int v \frac{du}{dx} dx$
A: They are the same. Just take note however which functions to take as $u$ and the other to take as $dv$.
According to K.A Straud in Engineering Mathematics, the function you take for $u$ should be differentiated to get $du$ while the one you take as $dv$ should be integrated to have $v$. He gave a priority of taking $u$ which is thus:


*

*$\ln x$

*$x^n$

*$e^x$

*Trigonometric functions.
After, the choice of $u$ and $dv$, you can then solve using any of the formulae. I hope you understand this?
A: The by-parts rule comes from the differentiation of a product:
$$(uv)'=u'v+uv'$$ so that
$$\int(uv)'\,dx=uv=\int u'v\,dx+\int uv'\,dx$$ and
$$\int u'v\,dx=uv-\int uv'\,dx.$$
If you denote $u'$ as $w$, you have $u=\displaystyle\int w\,dx$ and
$$\int wv\,dx=\left(\int w\,dx\right)v-\int \left(\int w\,dx\right)v'\,dx.$$
