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Integration by Parts Problems where You Go around in Circles

Sometimes if you use integration by parts twice, you get back to where you started from — which, unlike getting lost, is not a waste of time.

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First, use the LIATE mnemonic device to pick your u:

image1.png

To pick your u, go down this list in order; the first type of function on this list that appears in the integrand is the u.

image2.png

Now fast forward to the formula step:

image3.png

Now, substitute the right side of the above equation for the

image4.png

from the original solution:

image5.png

You can now solve this equation for the integral

image6.png

Use I in place of that integral to make this messy equation easier on the eyes:

image7.png

Add I to both sides:

image8.png

Multiply both sides by 1/2:

image9.png
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