Principale value, how can we consider it? How can we consider principal value where as, it's something that "doesn't exist". For example,
$$\int_{-\infty }^\infty \frac{1}{x}\mathrm d x$$
doesn't exist, but the principal value is nulle. What is the motivation between the principale value ?
Let $f:\mathbb R^2\longrightarrow \mathbb R$ a function. You can have that $$\lim_{(x,y)\to (0,0)}f(x,y)$$
doesn't exist, but that $$\lim_{x\to 0}f(x,g(x))$$
exist for some function $g:\mathbb R\to \mathbb R$ such that $$g(x)\underset{x\to 0}{\longrightarrow }0.$$

Now, the (Cauchy) principale value is the same idea. Indeed, let says that $x&lt;0$ and $y&gt;0$ and $$f(x,y)=\int_{-\infty }^x g(t)\mathrm d t+\int_{y}^\infty g(t)\mathrm d t.$$
Then $$\lim_{(x,y)\to 0}f(x,y)$$
may not exist, but we can get interested to $$\lim_{x\to 0}f(-x,x)$$
that is well defined if it exist. In other word, you define a function $$h(x)=f(-x,x),$$
and compute $$\lim_{h\to 0}h(x).$$
Sice $\frac{1}{x}$ is an odd function, the value of the integral must be zero. An for calculation you can calculate it in this way
$$I=\lim_{a\to \infty}\int _{-a}^{a} \frac{1}{x}dx=0$$

Hope this will be helpful !