IB Mathematics AA HL Flashcards -Integration by parts

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[h] IB Mathematics AA HL Flashcards -Integration by parts

[q] INTEGRATION BY PARTS

[a]

This is a method that can be derived by the product rule. If we remember that the rule states: \( (fg)’ = f’g + fg’ \), then:

\[
\int(fg)’dx = \int(f’g + fg’)dx
\]
\[
= fg – \int f’g dx
\]

This is not very useful in this form, but if we rearrange slightly, then we get a formula that helps us integrate most products of functions:

\[
\int f g’ dx = fg – \int fg dx
\]

[q]

We can tidy this up by substituting:
\[
u = f(x), \quad v = g(x), \quad du = f'(x)dx, \quad dv = g'(x)dx
\]

which simplifies to:

\[
\int u dv = uv – \int v du
\]

So as long as we have a \( g'(x) \) that we can integrate to \( g(x) \), then this will work. It is advisable to write out \( fg \), then \( f’ g \), then plug all into the formula.

MULTIPLE USES
Sometimes, you fill in the \( f’ g \) part, and that itself will need integration by parts to complete. Make sure you keep everything very organized.

 
 

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IB Mathematics AA HL Flashcards – Integration by parts

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