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I'm in differential equations without having taken higher level calculus first

I understand differentiation and integration pretty well but I get surprised calculus-wise periodically.

For example, how is it that $\int{e^{xt}dt} = \frac{e^{xt}}{x} + c$? I must be missing something!

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    same as $\int e^{3t}dt=e^{3t}/3$2011-03-29

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The definition of $e^x$ is:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

Then the derivative of $e^x$ is

$\frac{d}{dx} e^x = 0 + 1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots = e^x$

by the power rule.

You get your identity by applying the chain rule.

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Let $u=xt$ so that $t=\frac{u}{x}$ and $dt=\frac{du}{x}$. $\int e^{xt}dt =\int \left(e^u\cdot\frac{du}{x}\right) =\frac{1}{x}\int e^u du =\frac{1}{x}e^u+C =\frac{e^{xt}}{x}+C$

(Note: the integral you started with is with respect to $t$; as far as the integral is concerned, we're treating $x$ as a constant.)

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    Oh! Of course, thanks. Your U substitution made me realize what I was missing in the first place.2011-03-28
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Do you know that $\int e^u du = e^u +C$

Now think of integration techniques. Does substitution ring a bell?

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    Yeah, now I realize that I was completely missing the fact that I needed $x$/x to give me the x I needed for the integration, leaving 1/x outside the integral. Thanks2011-03-28