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!

Advertisements

Author: joelkindiak

Build people up. Point them to Christ.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s