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Most of the integration problems are bounded with integration rules. The formulas are only methods to do a integration problem. But trigonometry chapters like multiple angles or sub multiple angle are very impotent to simplify the integration as well. If you find a integration problem which is hard just try to think totally different way every big and complex integration are short in between 4-5 steps. And if its taking long then check again.

Mind it In complex integration  we often use substitution methods.
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Integration is smthng which has more of practical applicatons rather tha simply analytical...so always try to have  a practical approach towards it...and most imp...approach any problem systematically...formula r used at the end...first try to simplify as much as u can...i hope u will develop these skills
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