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