Evaluación de la integralidad $\int_{0}^1 \frac{\arctan x }{1+x} dx$

Question

¿Cómo se calcula $$\int_{0}^1 \frac{\arctan x }{1+x} dx$$

Answer 1

Utilizar la integración por partes $$\int_0^1 \frac{\arctan x}{1+x} dx = \arctan(x) \ln(1+x)|_0^1 - \int_0^1 \frac{\ln (1+x)}{1+x^2}dx$$

La primera parte es $\displaystyle \frac{\pi}{ 4} \ln 2$ y la última parte es $\displaystyle \frac{\pi}{8} \ln 2$ que se responde aquí .

Answer 2

$$\begin{align} \int_0^1\frac{\arctan(x)}{1+x}\,\mathrm{d}x &=\int_0^{\pi/4}\frac{\theta}{1+\tan(\theta)}\,\sec^2(\theta)\,\mathrm{d}\theta\tag{1}\\[6pt] &=\int_0^{\pi/4}\frac{\theta\,\mathrm{d}\theta}{\cos(\theta)\,(\cos(\theta)+\sin(\theta))}\tag{2}\\[6pt] &=\int_0^{\pi/4}\frac{(\frac\pi4-\theta)\,\mathrm{d}\theta}{\cos(\theta)\,(\cos(\theta)+\sin(\theta))}\tag{3}\\[6pt] &=\frac\pi8\int_0^{\pi/4}\frac{\mathrm{d}\theta}{\cos(\theta)\,(\cos(\theta)+\sin(\theta))}\tag{4}\\[6pt] &=\frac\pi8\int_0^{\pi/4}\frac{\sec^2(\theta)}{1+\tan(\theta)}\,\mathrm{d}\theta\tag{5}\\[6pt] &=\frac\pi8\int_0^1\frac1{1+x}\mathrm{d}x\tag{6}\\[6pt] &=\frac\pi8\Big[\log(1+x)\Big]_0^1\tag{7}\\[12pt] &=\frac\pi8\log(2)\tag{8} \end{align}$$ $(1):$ $x=\tan(\theta)$

$(3):$ $\theta\mapsto\frac\pi4-\theta$

$(4):$ desde $(2)=(3)$ tenemos $(3)=\frac{(2)+(3)}{2}$

$(6):$ $x=\tan(\theta)$