# By Parts Backstory

Here’s the simple concept behind integration by parts as described in an earlier post. Integration by Parts simply what I call the reverse-product rule. Here’s why.

Let $u$ and $v$ be functions of $x$. Using product rule, when we differentiate $uv$, we differentiate the first term *times* keep the second constant *plus* differentiate the second term *times* keep the first constant. That is,

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

Notationally, since $u'=\displaystyle\frac{du}{dx}$ and $v'=\displaystyle\frac{dv}{dx}$, we rewrite the above identity as

$\displaystyle \frac{d}{dx} \left(uv\right)=u'v + uv'$.

Let’s integrate with respect to $x$ on both sides. On the LHS, we are integrating what we get after differentiating. Since they cancel each other out, we get $uv$ on the LHS. On the RHS, we get the integrals of each chunk added together, that is,

$\displaystyle uv = \int u'v\ dx + \int uv'\ dx$.

Subtracting by $\displaystyle \int u'v\ dx$ on both sides and switching the LHS with the RHS, we get

$\displaystyle \int uv' \ dx = uv - \int u'v\ dx$.

which is the famous integration by parts formula.

Hope this was insightful on how this technique is nothing more than reversing the product rule!