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I believe that to be good at integration and maths in general practice is needed. Enough practice can give the student enough knowledge to be able to solve any question. For example, integration by parts involving trigonometry might be tricky, but with enough practice on integrating the basics and slowly increasing the difficulty will enable the student to solve hard questions using basic integration principles. Often students (including me) think they can skip steps when doing integration by parts, but it is important that you clearly write the steps out so that you don't make any silly mistakes such as a wrong negative sign or errors involving powers.