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One reason it is important to be able to identify a proper rational fraction is that these are the only rational fractions that can be integrated using the method of partial fractions. If you come across an improper rational fraction, you will first need to rewrite it as a proper rational ...
In summary, when approaching an integral with no idea how to solve it, it's important to first identify the type of integral and then use techniques such as substitution, integration by parts, or trigonometric identities. If stuck, take a step back, try a different approach, ...