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Suppose we have two integers $a$ and $b$.

What's the fastest way to get an exact value for $\int_a^b{(1+x)^n dx}$, with $n$ large?

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I suppose asking Wolfram Alpha is faster than asking here. Or do you want something other than the formula from the Fundamental Theorem of Calculus? – GEdgar Feb 19 '12 at 14:56
@GEdgar: I was just kind of wondering, because I am considering what I think is a neat trick to doing some integrals. I wasn't anticipating David Mitra's answer, though. My idea might not be as worthwhile as I thought it was. – Matt Groff Feb 19 '12 at 15:01
up vote 4 down vote accepted

Use a substitution, $u=1+x$: for $n\ne -1$ $$ \int_a^ b(1+x)^n\,dx=\int_{1+a}^{1+b} u^n \,du={u^{n+1}\over n+1}\biggl|_{1+a}^{1+b} ={(1+b)^{n+1}\over n+1} -{(1+a)^{n+1}\over n+1} . $$

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It took 6 minutes to the answer by asking here. How long by asking Wolfram Alpha? The question was "the fastest way"... – GEdgar Feb 19 '12 at 18:23
Plus, he said "$n$ large", so $n \ne -1$. – GEdgar Feb 19 '12 at 18:23

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