Kiana Dodson

2022-06-29

Antiderivative of $\mathrm{arctan}\left(-{x}^{2}\right)$.
As I said in the title I'm trying to find an antiderivative of $f\left(x\right)=\mathrm{arctan}\left(-{x}^{2}\right)$.
I am aware that e.g. WolframAlpha can find one, but I have no clue how to do it by hand. Can anyone give me a hint?

britspears523jp

Expert

Step 1
If you integrate by parts, you get $\int \mathrm{arctan}\left(-{x}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}dx=x\mathrm{arctan}\left(-{x}^{2}\right)-\int x\frac{-2x}{1+{x}^{4}}\phantom{\rule{thinmathspace}{0ex}}dx$
Step 2
Now find the partial fraction decomposition $\frac{2{x}^{2}}{1+{x}^{4}}=\frac{Ax+B}{{x}^{2}+x\sqrt{2}+1}+\frac{Cx+D}{{x}^{2}-x\sqrt{2}+1}$ and the rest is standard.

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