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 , we get the following:

which can be rearranged to get the product rule

Or in function notation

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

Higher order derivatives

It follows

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

Curl

5. Scaling

Semi-Riemannian geometry

To covariantly integrate by parts we can use .

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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