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.

---
### 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.

This guide provides a detailed step-by-step solution for integrating 1/(logx)^2 using substitution and integration by parts. The solution introduces concepts such as the exponential integral function and logarithmic properties, suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation