My weakest point in calculus has always been solving actual integrals. Of course I understand how everything works, and can apply things. The problem keeps that I waste a lot of time on, as discovered later, dead ends...
I have a lot of trouble deciding if I see a random function how I would start. Should I try to use the integration by parts rule? The substitution rule (good substition)? Convert the function first to a different function? Especially when things become non trivial and a combination is required I more often than not first try the wrong method.
Is this problem just a lack of experience and something that can only be solved by exercising more problems. Or are there tricks/standard approaches (ie: first try substitution then ... ) I can follow?
An very simple example of a solution I simply couldn't see is (though this is just an example):
$$\int {\frac{1}{x^2+6x+13}}dx$$
Answer
Part is obviously experience which allows you to more easily recognise what kinds of integrals you're dealing with but a large part is simply understanding the different methods of integration and why they work. This should make it easier for you to "diagnose" problems.
e.g. Integration by substitution often works well when an integral has a similar form to one that you already know (for instance your example and $\int \frac{1}{x^2+1} \mathrm{d}x=\arctan(x)$). Integration by parts often works well when you can identify two multiplying functions, one of which isn't too costly to integrate and the other which differentiates to zero eventually or in some cases to the same type of function. etc.
Of course throughout this process you also need to keep a keen eye out for any simplifications or other changes you might be able to make e.g. trigonometric identities, indices, etc.
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