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How do I solve this integral?

$$\int_0^1 \frac {x^4 (1-x)^4}{x^2+1} dx$$

emacs drives me nuts
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Mich Vaughan
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    See here & here. – J.G. Aug 19 '22 at 18:25
  • And in general, look into integration by partial fractions---it's the go-to method to integrate rational functions (and is especially useful when the denominator's factorization into linear and quadratic polynomials is "easy"). – user7530 Aug 19 '22 at 18:34
  • If you want some more insight into the workings of this integral, you may look at this answer (which is admittedly mine) and make sure to see some links I’ve posted in the comments. – insipidintegrator Aug 19 '22 at 18:45

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The first thing I would notice that this a "rational" function and that the numerator has higher degree than the denominator so we can divide to get a polynomial plus a rational function with denominator of higher degree than the numerator.

If I have done the algebra correctly $\frac{x^4(1- x)^4}{x^2+ 1}= x^6- 4x^5+ 5x^4- 4x^2+ 4+ \frac{-3}{x^2+ 1}$

It should be easy to integrate that.

George Ivey
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