Having trouble doing the integral of this problem, please help?

[twohaha]

\(\displaystyle \int x/(x^4+x^2+1)dx\)

The denominator can't be factored, so partial fractions don't work.
I can't seem to find an appropriate substitution. For example:

\(\displaystyle u=x^2 \)
\(\displaystyle du = 2xdx\)

\(\displaystyle 0.5 \int du/(u^2+u+1)dx\)

After completing the square, i get

\(\displaystyle 0.5 \int du/((u+1)^2 - u)dx\)

I can't find any other way to proceed after this. Does anyone have any suggestions?

Thanks!

 

[Deleted member 4993]

\(\displaystyle \int x/(x^4+x^2+1)dx\)

The denominator can't be factored, so partial fractions don't work.
I can't seem to find an appropriate substitution. For example:

\(\displaystyle u=x^2 \)
\(\displaystyle du = 2xdx\)

\(\displaystyle 0.5 \int du/(u^2+u+1)dx\)

After completing the square, i get

\(\displaystyle 0.5 \int du/((u+1)^2 - u)dx\)

I can't find any other way to proceed after this. Does anyone have any suggestions?

Thanks!

Complete the square such that you have only constants out side the square sign.

u² + u + 1 = (u + ½)² + ¾

Now you have familiar [tan^-1] form for integral.