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!


Author: joelkindiak

Build people up. Point them to Christ.

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