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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$\begingroup$ 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? $\endgroup$– GEdgarCommented Feb 19, 2012 at 14:56
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$\begingroup$ @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. $\endgroup$– Matt GroffCommented Feb 19, 2012 at 15:01
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1 Answer
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Use a substitution, $u=1+x$: for $n\ne -1$ $$ \int_a^ b(1+x)^n\,dx=\int_{1+a}^{1+b} u^n \,du={u^{n+1}\over n+1}\biggl|_{1+a}^{1+b} ={(1+b)^{n+1}\over n+1} -{(1+a)^{n+1}\over n+1} . $$
answered Feb 19, 2012 at 14:52