(Sorry DeMath, if this is far from what you had in mind! But anyway...)

Just in case a picture helps...

(As usual, straight lines differentiate downwards or integrate up.)

The essence of integration by parts is to work backwards through the product rule for differentiation...

... we want to fill out this product-rule shape usefully, but starting in one of the bottom corners instead of - as happens in differentiation, the top one. Choosing the right arrangement can be tricky, but once you've started in the right way, as above, it's really a matter of following through. Fill in the rest of the product-rule shape, then you'll need to cancel the addition you just entered, to make the lower equals sign valid, again.

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Don't integrate - balloontegrate!

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