To solve the integral of $\frac{1}{(\log{x})^2}$ with respect to $x$, we can apply a substitution method.

Step-by-Step Solution:

1. Substitution: Let $u = \log{x}$.
   Therefore, $du = \frac{1}{x} dx$, which gives us $dx = x \, du$.

   Since $u = \log{x}$, we also have $x = e^u$. Thus: $dx = e^u \, du$

2. Rewrite the Integral: Substituting these into the integral, we get: $\int \frac{1}{(\log{x})^2} \, dx = \int \frac{1}{u^2} \cdot e^u \, du$

3. Integrating by Parts: Now, we integrate $\frac{e^u}{u^2}$ using integration by parts.

   Let $v = \frac{1}{u^2}$ and $dw = e^u du$.

   So, we apply integration by parts: $\int v \, dw = v \cdot w - \int w \, dv$ This method simplifies the integral, yielding: $\int \frac{e^u}{u^2} \, du = -\frac{e^u}{u} + \text{Ei}(u)$ where $\text{Ei}(u)$ is the exponential integral function.

4. Substitute Back: Substituting back $u = \log{x}$, we get: $\int \frac{1}{(\log{x})^2} \, dx = -\frac{1}{\log{x}} + \text{Ei}(\log{x}) + C$ where $C$ is the constant of integration.

Final Answer:

$\boxed{-\frac{1}{\log{x}} + \text{Ei}(\log{x}) + C}$

Let me know if you'd like any clarification or additional details.

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Related Questions:

1. How can we use the substitution method for integrals with different logarithmic expressions?
2. What is the definition and properties of the Exponential Integral function $\text{Ei}(x)$?
3. Can we solve $\int \frac{1}{(\log{x})^n} \, dx$ for other values of $n$?
4. How do we approach integrals involving products of exponentials and powers of logarithms?
5. What are the applications of logarithmic integrals in calculus?

Tip:

When solving integrals involving logarithms, substitution is often useful to simplify the expression, especially when combined with integration by parts.