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I haven't studied multiple integrals yet, but when I look at the integration by parts formula, I see that there is an integral of an integral. If the integration being done has limits, I'm not sure how they're managed. Here is what I have:

So the by parts formula is \int f(x)g'(x)dx = f(x)g(x)-\int f'(x)g(x)dx

If I have g'(x), then I need $g(x)$.

My question is this: if I'm solving with limits, do I apply limits to \int g'(x)dx also?

In other words, do I do this

$\int_a^b xe^{6x}dx = x \cdot |\frac{1}{6}e^{6x}|_a^b - \int_a^b1\cdot \frac{1}{6}e^{6x}dx$

or this

$\int_a^b xe^{6x}dx = x \cdot |\frac{1}{6}e^{6x}|_a^b - \int_a^b1\cdot |\frac{1}{6}e^{6x}|_a^bdx$

?

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    @joriki, yes.. Seems I'm dyslexic today :)2012-02-08

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It goes like this:

$\int_a^b xe^{6x}dx = (x \cdot \frac{1}{6}e^{6x})|_a^b - \int_a^b(1\cdot \frac{1}{6}e^{6x})dx$

$f(x) g(x)$ is evaluated from $a$ to $b$ and so is the integral \int^b_a [f'(x) g(x)] dx