Resol I _ { 3 } = ∫ (from 0 to frac { pi) of {4}}f(tanx) wrt x

$$I _ { 3 } = \int _ { 0 } ^ { \frac { \pi } { 4 } } f ( \tan x ) d x$$

Resoleu I_3

$$I_{3}=\frac{\ln(2)f}{2}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}$$

Resoleu f

$$\left\{\begin{matrix}f=\frac{4I_{3}\cot(x)}{\pi }\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }\left(x>\frac{\pi n_{2}}{2}\text{ and }x<\frac{\pi n_{2}}{2}+\frac{\pi }{2}\right)\\f\in \mathrm{R}\text{, }&I_{3}=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\end{matrix}\right,$$