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How to calculate this difficult integral: $\int\frac{x^2}{\sqrt{1+x^2}}dx$?

The answer is $\frac{x}{2}\sqrt{x^2\pm{a^2}}\mp\frac{a^2}{2}\log(x+\sqrt{x^2\pm{a^2}})$.

And how about $\int\frac{x^3}{\sqrt{1+x^2}}dx$?

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    WA now seems to understand LaTeX and so it is easy to experiment and see a general form. Try http://www.wolframalpha.com/input/?i=%5Cint%5Cfrac%7Bx%5E3%7D%7B%5Csqrt%7B1%2Bx%5E2%7D%7Ddx and change the exponent.2011-09-14
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    Where did $a$ come from?2011-09-14
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    There is a nice recursion you can derive: letting $\mu_n=\int \frac{t^n}{\sqrt{t^2+1}}\mathrm dt$, we have $$\mu_n=\frac1{n}(t^{n-1}\sqrt{1+t^2}-(n-1)\mu_{n-2})$$. The integrals for $n=0,1$ are easily derived, so you can use those to start the recursion.2011-09-14
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    but how did you get this recursion?2011-09-14

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