# Tag Archives: arctan

## Integral of arctan and log using contour integration

$$\int^\infty_0\frac{\log\left(x^2+1 \right)\arctan^2\left(x\right)}{x^2}\,dx = \frac{\pi^3}{12}+\pi \log^2(2)$$ Lemma $$\int^\infty_0 \frac{\log^3(1 + x^2)}{x^2}\,dx = \pi^3+ 3 \pi \log^2(4)$$ Start by the following $$\int^{\infty}_0 x^{-p}(1+x)^{s-1} dx= \frac{\Gamma(1-p)\Gamma(p-s)}{\Gamma(1-s)}$$ Let $x\to x^2$ $$\int^{\infty}_0 x^{-2p+1}(1+x^2)^{s-1} dx= \frac{\Gamma(1-p)\Gamma(p-s)}{2\Gamma(1-s)}$$ Let $p = 3/2$ \int^{\infty}_0 \frac{1}{x^2(1+x^2)^{1-s}} … Continue reading

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