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Compute $\int_{-2}^6 (8x^5-6x^3+7x-3-22\sin(\pi x))\,dx$.

Simple question: When I take the indefinite integral of this in $0.1$ seconds, is there a way to avoid spending the next $2$ hours crunching through arithmetic?

Saketh Malyala
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  • Except that $x\mapsto\sin(\pi x)$ is $2-$periodic, with zero meanvalue, which implies that $\int_{-2}^6\sin(\pi x),dx=0$, I don't see any fast way to compute the other terms, but it shouldn't take more than 2 mins. – Adren Mar 10 '17 at 05:16

2 Answers2

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You could recognize that the integral of $\sin$ over a full period $(2\pi)$ is zero and ignore that term. Computing $\frac 86(6^6)$ without a calculator is not so easy. If you know $6^3=216$ you can square that but what you want is $6^5=7776$. Yes, I know that one from computing so many probability problems involving dice. It shouldn't take two hours by hand.

Ross Millikan
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The short answer is no. However a trick you can use is simplifying each term one at a time. For example, $\int_{-2}^{6}8x^5\text{d}x$ first to get $\frac{4x^6}{3}\big\rvert_{-2}^{6}$ and then do it like that for the rest of the polynomial terms. Another way is using simpsons rule since it will be exact because of the degree of 3, leaving out the $x^5$ term and doing that seperately. So taking things term by term like this you can avoid the tedious simplification of doing say $\frac{4}{3}(6^5)-\frac{3}{2}6^4+\frac{7}{2}(6^2)...$ and then minus the stuff at negative 2. Also this will take a while either way but doing it term by term reduces the work. Of course at the end you will still have to combine all the terms

Teh Rod
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