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I have this integral to evaluate: $\int e^{x}(1-e^x)(1+e^x)^{10} dx$

I figured to use u substitution for the part that is raised to the tenth power. After doing this the $e^x$ is canceled out.

I am not sure where to go from here however due to the $(1-e^x)$.

Is it possible to move it to the outside like this and continue from here with evaluating the integral?

$(1-e^x)\int u^{10} du$

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    No, we cannot move it outside, it is not a constant. Let $u=1+e^x$. Then $du=e^x\,dx$ and since $e^x=u-1$, we find that $1-e^x=1-(u-1)=2-u$. – 2011-12-22

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If $u=1+e^x$, then $e^x=u-1$, so $1-e^x=1-(u-1)=2-u$. Now you can easily multiply out. Note that the ‘bad’ factor could be any simple polynomial in $e^x$, and the technique would still work. For instance, if the integrand had been $e^x(3e^{2x}-4e^x+5)(1+e^x)^{10}$, substituting $u=1+e^x$ would turn it into $(3u^2-2u+4)u^{10}$, since

$\begin{align*} 3e^{2x}-4e^x+5&=3(u-1)^2+4(u-1)+5\\ &=3u^2-2u+4\;. \end{align*}$

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Let $u=e^x$, $du=e^xdx$, so your integral is

$\int (1-u)(1+u)^{10}du.$

Now there are many ways for you to proceed (the quickest probably being integration by parts).

Addendum: to prove my claim that the quickest way is by parts, here it is:

$\int (1-u)(1+u)^{10}du = \int(1-u)d\left(\frac{(1+u)^{11}}{11}\right) =$

$(1-u)\left(\frac{(1+u)^{11}}{11}\right) + \int \frac{(1+u)^{11}}{11} du = \frac{(1-u)(1+u)^{11}}{11}+\frac{(1+u)^{12}}{132}+C.$

I challenge anyone (cough! cough! :-)) to do it quicker by expanding $(1+u)^{10}$.

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    @Srivatsan, thanks for sharing, I LOL'ed! – 2011-12-22
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let $x=\ln(u)$

$dx=du/u$

$I=\int e^{x}(1-e^x)(1+e^x)^{10} dx$ = $\int ((u(1-u)(1+u)^{10})/u)du$=$\int (1-u)(1+u)^{10}du$

You may want to take it from here...

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FWIFs, this also would be easily generalized via a recurrence relation (aside from the obvious substitution that confirms this] as the integral is of the form;

f ' (x) times g(x) (where f(x) is (1/11)(1+e^x)^11) and f(x), g(x) both behave good enough - i.e. we get back something close enough to our integral the define a recurrence.