# Evaluating $\int \frac{5-e^{x}}{e^{2x}}dx$ without Partial Fractions

I am trying to evaluate

$$\int \frac{5-e^{x}}{e^{2x}} \mathrm dx$$

I tried rewriting the integral by throwing $e^{2x}$ up on top and using $$u=e^{x}$$ $$du = e^{x} dx$$

I then tried another substitution where $v = 5-u$ and $dv = -1 du$ but then I can only simplify the integral to

$$\int \frac{v}{(v-5)^{3}} \mathrm dv$$

Which would then require partial fractions, which my class has not gotten to quite yet (so I'm not allowed to use the method for homework, sadly).

Is there a simple substitution I am overlooking from the beginning or something?

Thanks.

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Your substitution would work, but the approaches discussed in the answers are simpler. For example, let $u=e^x$ as you did. Then your integral becomes $\int \frac{(5-u)}{u^3}\,du$. This is $\int \left(\frac{5}{u^3}-\frac{1}{u^2}\right)du$, which is very doable without partial fractions. –  André Nicolas Mar 13 '12 at 4:39
Ah, I see that now. For some reason my mind wasn't in a mode to split up the integrand into two pieces last night. –  Joe Mar 13 '12 at 21:02

This is (homework), so only hints: $$\begin{eqnarray} \frac{a-b}{c} & = & \frac{a}{c} - \frac{b}{c} \\ \frac{e^{\alpha x}}{e^{\beta x}} & = & ?? \\ \int e^{tx} \ dx & = & ?? \\ \end{eqnarray}$$

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Ah, wow. I was simply overthinking the problem! Thanks J.D. –  Joe Mar 13 '12 at 4:09
Do you have to use the quotient rule in order to finish with this problem? –  Victor Mar 13 '12 at 6:10
@Victor: The quotient rule is for differentiation, not integration. The middle equation here tells us that the fractions can be simplified. –  anon Mar 13 '12 at 9:30

Rewrite the integrand $5e^{-2x}+e^{-x}$. You know how to integrate $ae^{bx}$ right?

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Ah, of course. I glossed over that fact (it's late, that's an excuse, right?) - would much rather keep it simple than double substitutions and partial fractions. Thanks. –  Joe Mar 13 '12 at 4:10

Did you try integrating by parts where $dv = \frac{3}{e^{3x}}$ and $u = 5 - (e^x)$ ? It may be a bit laborious of a calculation,but that's what I'd try if I was instructed NOT to use partial fractions. This is where knowing as many techniques of integration as possible comes in very handy in baby calculus.

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Cmon! Here's the LaTeX :) $dv = \frac{3}{e^{3x}}$ and $u = 5 - (e^x)$ –  user2468 Mar 13 '12 at 4:27
@J.D. That better? –  Mathemagician1234 Mar 13 '12 at 5:12
No, I did not try integrating by parts - my class is just learning antiderivatives of functions involving e, so I figured that method was also implied not to be used, as the class has not learned it. Nonetheless, I see how that would also work. Rewriting the integrand and splitting it up into two integrals seems to be the easiest way from what I could tell though. –  Joe Mar 13 '12 at 21:00
You would really use integration by parts rather than the suggestion in the earlier answers of distributing the division and integrating simple exponentials? –  Jonas Meyer Mar 14 '12 at 3:41
Yes, but it's a terrible way of doing the problem. Not only does it use unnecessarily technology, but the algebra is harder than the elementary solution given above. [moderation edited to remove superfluous insult, M.G.] –  Adam Smith Mar 14 '12 at 6:38