## Integration by Parts

This is for my homies taking Calculus.

U-substitution is the opposite of the Chain Rule.  Integration by parts is the opposite of the Product Rule.

Don’t believe me?

Take u and v to be the left and right parts of some formula.

$\dpi{200} \bg_white \begin{matrix} [ \ u \cdot v \ ] \; ' &= \ \ \, u \; ' \cdot v \ &+ \ \ u \cdot v \; ' \qquad &\text{(product rule)} \\ \\ [ \ u \cdot v \ ] \; ' &- \ \ u \; ' \cdot v \, \ &= \quad u \cdot v \; ' \qquad &\text{(algebra)} \\ \\ \smallint [ \ u \cdot v \ ] \; ' \ &- \ \smallint u \, ' \cdot v \ \ &= \ \smallint u \cdot v \; ' \qquad &( \smallint \text{ int)} \\ \\ \ u \cdot v \quad &- \; \smallint du \cdot v \ \ &= \, \smallint u \cdot dv \qquad &\text{(parts formula)} \end{matrix}$$\dpi{200} \bg_white \begin{matrix} [ \ u \cdot v \ ] \; ' \quad &= \ \ \, u \; ' \cdot v \ + \ u \cdot v \; ' \qquad &\text{(product rule)} \\ \\ [ \ u \cdot v \ ] \; ' &- \quad u \; ' \cdot v \, \ &= \quad u \cdot v \; ' \qquad &\text{(algebra)} \\ \\ \smallint [ \ u \cdot v \ ] \; ' \ &- \ \smallint u \, ' \cdot v \ \ &= \ \smallint u \cdot v \; ' \qquad &\text{(int)} \\ \\ \ u \cdot v \quad &- \; \smallint du \cdot v \ \ &= \, \smallint u \cdot dv \qquad &\text{(parts formula)} \end{matrix}$

Now switch some symbols around and you’ve arrived at the formula for Integration by Parts.

By the way, I normally use L and R for the left and right groups of symbols when I’m teaching Product Rule.  Here I just used u and v because that’s probably how you’ve seen the formula be written.

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