Answer. We want to choose u so that u' is simpler than u. This is probably a good choice:
u = x
v' = sin x
Now u' = 1 is simpler than u and v = (-cos x) isn't any worse than v'.
We apply the formula:
Simplifying the right-hand side and evaluating the new integral gives us
We could also write this as
It's possible that after applying the formula for integration by parts, you'll need to use integration by parts again in order to figure out the new integral. If that happens, be very careful with signs and coefficients. We recommend you work out the new integral by itself, then wrap up your expression for the new integral in parentheses before putting it back into the formula.
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