Integration techniques MOC

Integration by parts

Integration by parts is basically an integral version of the product rule.¹² By beginning with the product rule, and integrating by 𝑥, we get the following:

𝑑𝑑𝑥(𝑢𝑣)=𝑣𝑑𝑢𝑑𝑥+𝑢𝑑𝑣𝑑𝑥⟹𝑢𝑣=∫𝑣𝑑𝑢+∫𝑢𝑑𝑣

which can be rearranged to get the product rule

∫𝑢𝑑𝑣=𝑢𝑣−∫𝑣𝑑𝑢

Or in function notation

∫𝑓(𝑥)𝑔′(𝑥)𝑑𝑥=𝑓(𝑥)𝑔(𝑥)−∫𝑔(𝑥)𝑓′(𝑥)𝑑𝑥

It applies to the situation in which you are called upon to integrate the product of one function (𝑓) and the derivative of another (𝑔); it says you can transfer the derivative from 𝑔 to 𝑓, at the cost of a minus sign and a boundary term.

Higher order derivatives

It follows

∫𝑏𝑎𝑑𝑛𝑓𝑑𝑥𝑛𝑔𝑑𝑥=[𝑛∑𝑗=1𝑑𝑛−𝑗𝑓𝑑𝑥𝑛−𝑗𝑑𝑗−1𝑔𝑑𝑥𝑗−1]𝑥=𝑏𝑥=𝑎+(−1)𝑛∫𝑏𝑎𝑓𝑑𝑛𝑔𝑑𝑥𝑛𝑑𝑥

Generalizations

By exploiting different generalisations of the Fundamental theorem of calculus along with different generalisations of the product rule, we can generalise integration by parts for vector valued functions.³

Divergence

3. Scaling

∭Ω𝑓(⃗∇⋅⃗𝐆)𝑑𝜏=⊂⊃∬𝜕Ω𝑓⃗𝐆⋅𝑑⃗𝐚−∭Ω⃗𝐆⋅(⃗∇𝑓)𝑑𝜏

Curl

5. Scaling

∭Ω𝑓(⃗∇×⃗𝐆)𝑑𝜏=−⊂⊃∬𝜕Ω𝑓⃗𝐆×𝑑⃗𝐚−∭Ω(⃗∇𝑓)×⃗𝐆𝑑𝜏

Proof of 5

First note that according to ^GE5

𝑓(⃗∇×⃗𝐆)=⃗∇×(𝑓⃗𝐆)−(⃗∇𝑓)×⃗𝐆

whence applying ^C1 we have

∭Ω𝑓(⃗∇×⃗𝐆)𝑑𝜏=−⊂⊃∬𝜕Ω𝑓⃗𝐆×𝑑⃗𝐚−∭Ω(⃗∇𝑓)×⃗𝐆𝑑𝜏

proving ^GE5.

Semi-Riemannian geometry

To covariantly integrate by parts we can use ^P1.

Practice problems

• 2016. Calculus, pp. 516–518 (§7.1 exercises)
• 2022. MATH1011: Multivariable calculus, pp. 94–95 (examples 6.24–6.25)

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Footnotes

1. 2022. MATH1011: Multivariable calculus, pp. 94–95 (§6.3.6) ↩

2. 2016. Calculus, pp. 512–516 (§7.1) ↩

3. 2013. Introduction to electrodynamics, p. 37 (eqn 1.59) ↩