Integration by parts works if u is absolutely continuous and the function designated v′ is Lebesgue integrable (but not necessarily continuous). (If v′ has a point of discontinuity then its antiderivative v may not have a derivative at that point.) and so long as the two terms on the right-hand side are finite.
Can you always use integration by parts?
Integration by parts is for functions that can be written as the product of another function and a third function's derivative. A good rule of thumb to follow would be to try u-substitution first, and then if you cannot reformulate your function into the correct form, try integration by parts.
When can you not use integration by parts?
Whenever you're faced with integrating the product of functions, consider variable substitution before you think about integration by parts. For example, x cos (x2) is a job for variable substitution, not integration by parts. ... (If you prefer, you can also use the mnemonic Lousy Integrals Are Terrible.)
