ok its
(integral) sin x cos x dx
i did u= sinx dv= cosx du=cos x v=sinx
i then plug in and get sin^2 x - (integral)(sin x cos x)
that second term is exactly the problem we started with, so how does that help me?--it seems like i will just be doing an endless integration by continuing when that is clearly not what the answer dictates...
It's worth knowing, though (edit: and this is just what Soroban is also pointing out), that when the 'endlessness' thing occurs in integration by parts, you can often break out of it like this...
... i.e. by solving the top row for I.
... is the basic pattern for integration by parts, and...
... is the product rule, straight continuous lines differentiating downwards (integrating up) with respect to x, and choosing legs crossed or un-crossed is equivalent to all the labelling with u and v and du and dv.
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Don't integrate - balloontegrate!
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