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:

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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.

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Now fast forward to the formula step:

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

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from the original solution:

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You can now solve this equation for the integral

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

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Add I to both sides:

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Multiply both sides by 1/2:

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