Integration by Parts: Reversing the Product Rule It’s really easy to reverse the power rule and surprisingly easy to reverse the product rule. Reversing the quotient rules seems like a shot in the dark…another derivative rule that’s actually not that hard to reverse, and is kind of cool to reverse, actually, is the product rule. Let’s see if we can do it: Problem: Starting with
d [u ⋅ v ] = u ⋅ v′ + v ⋅ u ′ , the product rule, derive the formula for Integration by dx
Parts.
Simplify a little to get the formula for integration by parts:
∫ u ⋅ v′ = u ⋅ v − ∫ v ⋅ u ′
Integration by Parts (IBP) is most commonly written in the form in the box below: Integration by Parts
∫ u dv = u ⋅ v − ∫ v du What you do: 1. Make a choice for u and the rest is dv. (Or pick dv and the rest is u.) 2. Do it! If it works, you picked correctly. If it doesn’t work, switch it up. Problem: Evaluate
∫ x ⋅ cos ( x ) dx .
Two obvious hints that are basically how I make my choices: • Pick u so that the derivative get’s simpler (or lower degree…or preferably so that the derivatives eventually vanish if you keep taking them repeatedly…I’m looking at you polynomials!) • You have to pick dv so that you can actually integrate it or you’ve defeated the entire purpose… Other people use the acronym L.I.A.T.E. for making the choice of u. This tells you to give preference for u: Logs, Inverse Trig, Algebraic, Trig functions, and Exponential. I don’t use this because I have trouble remembering it, honestly, but I don’t want to limit you. Calc AB Notes 26
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Problem: Evaluate
∫ x ⋅e
x
dx .
Sometimes integration by parts works in a scenario where it seems like it might not…there are two pretty standard examples of that—so we’re going to do them. Problem: Evaluate
∫ ln ( x ) dx .
It’s very worth your while to memorize the antiderivative of ln ( x ) if you haven’t already. Problem: Evaluate ∫ sin −1 ( x ) dx .
I can’t think of a reason to memorize the antiderivative of sin −1 ( x ) . Calc AB Notes 26
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Problem: Evaluate
∫x e
dx . Be really careful with parentheses.
Problem: Evaluate
∫x e
dx using the table method for “rapid repeated Integration by Parts.”
3 x
3 x
u=
∫x e
3 x
dv =
dx =
The table method works best (as in, works) when the repeated derivatives eventually vanish. The next problem is not an example of this concept…not sure why I did that, but I did. Problem: Evaluate
Calc AB Notes 26
∫ x ln ( x ) dx . 2
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Loops! Sometimes these things loop on themselves! Who saw that coming? It’s a kind of neat, kind of weird phenomenon. Problem: Evaluate
∫e
x
cos ( x ) dx .
If neither derivative is really simpler and neither goes away, I usually pick the function I’d rather find an antiderivative of because derivatives are just work, antiderivatives sometimes require inspiration.
And…of course…sometimes you just need to mix and match your strategies. These are my favorite and, eventually, if you get really good at antiderivatives, you’ll probably enjoy them too! Problem: Evaluate
∫x e
3 x2
Problem: Evaluate ∫ sin
Calc AB Notes 26
dx .
( x ) dx .
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Problem: Evaluate
∫ (sin x )
Problem: Evaluate
∫ x ⋅sin (
−1
2
dx .
)
x dx .
Integration by parts also shows up frequently on volume by shells problems, which kind of makes sense if you think about it since we’re always multiplying in an extra factor with x for those. Problem: Use shells to find the volume of the solid formed when the first quadrant region bounded by y = sin ( x ) , x = 0 , and x = π is rotated about the y-axis.
Calc AB Notes 26
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