## Integration by parts

How did loving the ground-up toenails of bisexuals get an interior designer to take up geology? Simple, he went from noting decor to what the core denotes by being into grated bi-parts.

I don’t really get why this XKCD is funny.

But here is a picture explaining integration by parts:

The area of the entire rectangle is $uv$, and it is made of two parts we integrate, so

$uv = \int \!u\, \text{d}v + \int\! v\,\text{d}u$

and therefore

$\int \! u \,\text{d}v = uv - \int\! v \,\text{d}u$

Also, take $\text{d}(uv) = \text{d}(\int \!u \,\text{d}v + \int \!v\,\text{d}u)$ and you find

$\text{d}(uv) = u \,\text{d}v + v\,\text{d}u$,

which is the product rule.

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### 3 Responses to “Integration by parts”

1. Anonymous Says:

XKCD is vary rarely funny.

2. nicolas Says:

XKCD is vary rarely funny.

3. Ψε Says:

Nice visualisation! Always had this intuition, but couldn’t actually think of how simple this geometric intuition is.