there's no general key, it's practice... the only way to succeed is to make lots of problems and you realize what to do.
This is more of a general question, not targeted towards a specific problem. Well says you are given a general integral problem and it requires you to use substitution or integration by parts or both combined.
What are keys thing to look at it when doing either one or both in a problem ? I am trying to figure out the a logical process which I can understand rather than looking at the book endlessly
A couple of general strategies are at
Pauls Online Notes : Calculus II - Integration Strategy
and
http://www.math.psu.edu/dlittle/m141f06/strategy.pdf
Also, you may find it helpful to reflect on why differentiation is inherently more challenging done backwards than forwards. And, basically, that's because the chain-rule...
... and product-rule...
... for differentiation are much more particular and complicated at the 'output' end of the process (the bottom rows here, if that helps) than at the input end (the top rows). So we're much less likely to be able to put an algebraic function through the process backwards, and if we can we're more likely to have to transform it first.
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Don't integrate - balloontegrate!
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