How can I recognise when to use a particular method for finding an integral?
Often the question will ask you to use a particular method to integrate something, but sometimes they will not and you will have to recognise for yourself the best way to do it. There is often more than one way to do it, but finding the quickest and most efficient way is a skill. For example if you were asked to find the integral ∫x2lnx dx, you should notice that there there are two distinct parts the x2 and lnx that can be easily integrated and differentiated separately. The easiest way to do this question is to do it by parts.However, sometimes there is no technique and the fastest way of doing it is by looking at it and thinking "what would differentiate to give that". for example, for ∫sinxcos2x dx you may be tempted to try and do it by parts as it can be split. However this would actually be extremely difficult and waste a lot of time. The easiest way to do this is to realise that it is the result of differentiating something similar to cos3x, but differentiating cos3x would give -3sinxcos2x + C, so we need to have a coefficient of -1/3 to make it work. (-1/3)cos3x + C is the answer. We have effectively used the reverse of the chain rule.I cannot stress how important this skill is in an exam, you could waste a lot of valuable time in an exam trying the wrong technique; good time management is essential for achieving a high grade in mathematics exams.
