8.1. Specify the integer closest to the larger root of the equation

$$
\operatorname{arctg}\left(\left(\frac{2 x}{7}+\frac{7}{8 x}\right)^{2}\right)-\operatorname{arctg}\left(\left(\frac{2 x}{7}-\frac{7}{8 x}\right)^{2}\right)=\frac{\pi}{4}
$$
Solution. Let $y=\left(\frac{2 x}{7}-\frac{7}{8 x}\right)^{2}$, then $\left(\frac{2 x}{7}+\frac{7}{8 x}\right)^{2}=y+1$ and the equation will take the form

$$
\operatorname{arctg}(y+1)-\operatorname{arctg} y=\frac{\pi}{4}
$$

Since $0 \leqslant \operatorname{arctg} y<\operatorname{arctg}(y+1)<\frac{\pi}{2}$, the last equation is equivalent to

$$
\operatorname{tg}(\operatorname{arctg}(y+1)-\operatorname{arctg} y)=1 \Longleftrightarrow \frac{1}{1+(y+1) y}=1 \Longleftrightarrow y=0 \Longleftrightarrow \frac{2 x}{7}=\frac{7}{8 x} \Longleftrightarrow|x|=\frac{7}{4}
$$

Answer: 2.