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"What is the difficulty when calculating the integral

$$\int_0^∞ \frac{6}{x^7+2x+1} dx $$

? Solve the integral with matlab's quad."

Sorry for my bad formatting, I'm just learning this formatting. But anyway I think the problem is the infinite bounds so we should chop off the bounds somewhere. Can you suggested how to trim the bounds and divide this integral into perhaps two integrals that we can handle?

I'm thinking of dividing the integral into two manageable integrals and then use matlab's quad.

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It's okay if your formatting is not so great. Just look at the edits and learn, I guess. People are always gonna help if you're trying. There's a FAQ section about how to format stuff though, you should check it out. –  Patrick Da Silva Sep 10 '12 at 7:51
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The infinite bound is the most cumbersome part. Maybe you can split the integral into $\int_0^1 + \int_1^{\infty}$, and then make change of variable $x \mapsto 1/x$ to turn the latter into an integral on$[0, 1]$. –  sos440 Sep 10 '12 at 7:56

1 Answer 1

up vote 6 down vote accepted

One could decompose this into an integral from $0$ to $1$ and an integral from $1$ to $\infty$ and use the change of variable $x\to1/z$ in the second part. Since $\mathrm dx=\mathrm dz/z^2$, this yields $$ \int_1^{+\infty}\frac{\mathrm dx}{x^7+2x+1}=\int_0^1\frac{z^5\mathrm dz}{z^7+2z^6+1} $$ and, finally, the integral to compute is $$ 6\int_0^1\left(\frac1{x^7+2x+1}+\frac{x^5}{x^7+2x^6+1}\right)\mathrm dx. $$

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