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I need help better understanding how, and why integration by changing the variable works (I've seen it's related to the derivative of a composite function $f(g(x))$), and generally tips and tricks, an intuitive understanding and exercises.

I sort of understand it, but when it comes to evaluate such an integral, I'm very sluggish and it seems harder than it should be.

LE: My mistake, I originally said integration by parts, but I meant variable change.

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It is actually more or less a restatement of the Product Rule of differentiation. As to sluggishness, it is a question of practice, after a while you will have seen most common "types." – André Nicolas Aug 14 '12 at 9:46
And any tips on which to choose as $f(x)$ and which as $g'(x)$ or something? – andreas.vitikan Aug 14 '12 at 9:56
As @André says, recognizing when you should be using integration by parts requires a fair bit of experience. Some people make use of a certain "ILATE" mnemonic, but I believe it is better for you to practice and get a feel for which integrals are expediently done by integration by parts. – J. M. Aug 14 '12 at 9:57
The "ILATE" mnemonic, as @J.M. mentioned is particularly helpful when you are first starting out but then later on, with practice, you will be able to figure out the way without any mnemonics as such. In fact, I think using the mnemonic, while helpful at the beginning, will hinder you intuitive development when it comes to integrating by parts! – dashdart Aug 14 '12 at 10:04
Our comments still apply, even with your edit: it takes some amount of practice to recognize which substitutions should be done to an integral. – J. M. Aug 14 '12 at 10:07

1 Answer 1

up vote 0 down vote accepted

Hint: How about you take a look here

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This is more properly a comment, at least in this form that does not have much elaboration. – J. M. Aug 14 '12 at 9:55
agreed to this comment. – Seyhmus Güngören Aug 14 '12 at 10:02

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