so that u' is simpler than u and v isn't any worse than v':
u' = x
v = sin x
Sticking everything in the formula, we get
Using integration by parts again, we find that
Wrap this up in parentheses, and put it back in the integration by parts formula where we left off:
We don't care about including the + C in the parentheses, since it doesn't matter if we end up with + C or – C.
If we hadn't done that, it would have been very easy to write
which is not the correct answer (the term x cos x has the wrong sign). This is the step we're talking about when we say "be careful with your signs and coefficients" and "wrap the expression for the new integral in parentheses before putting it back in the formula."