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I am having hard time understanding integration by substitution method so can I relay on integration by parts?

user3461957
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  • There is no universal method for carrying out integration. Here's a recent post for an integral where integration-by-parts is useless! [ http://math.stackexchange.com/questions/1371551/what-is-the-mistake-in-doing-integration-by-this-method ] There are quite a few functions where one of the two methods is more convenient than the other, others where only one is helpful, and still others where neither will do the job. (And that's for functions where we can write down the anti-derivative [apropos of avid19's answer below] .) – colormegone Aug 01 '15 at 06:29
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    I doubt that integration by parts could be used successfully on $2xe^{x^2}$, which by substitution is immediate. – André Nicolas Aug 01 '15 at 06:33
  • You could apply it recursively, and possibly get a series expansion... – Zach466920 Aug 01 '15 at 18:00

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If by "integrate" you mean "find an elementary anti derivative" then no. There are some functions that don't HAVE elementary anti derivatives such as $$\int e^{x^2} dx$$

You can't find an elementary anti derivative for $e^{x^2} $ (using ANY method) because there doesn't exist one.