https://file.aiquickdraw.com/mathbot/file/1725802898247a5gg42pq.jpg

The image shows an integral expression:

\[
\int \frac{1}{(x-a)(x-b)(x-c)} \, dx
\]

This integral can be solved using the method of partial fractions. Let's proceed with that.

### Steps for Solving the Integral:

1. **Partial Fraction Decomposition:**
   Decompose the given integrand into partial fractions:
   \[
   \frac{1}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}
   \]
   Here, \( A \), \( B \), and \( C \) are constants that need to be determined.

2. **Finding Constants:**
   Multiply both sides by the denominator \( (x-a)(x-b)(x-c) \) to get:
   \[
   1 = A(x-b)(x-c) + B(x-a)(x-c) + C(x-a)(x-b)
   \]
   Now, solve for \( A \), \( B \), and \( C \) by substituting suitable values for \( x \), such as \( x = a \), \( x = b \), and \( x = c \), or by comparing coefficients.

3. **Integrate Each Term:**
   After finding \( A \), \( B \), and \( C \), integrate each term separately:
   \[
   \int \frac{A}{x-a} \, dx + \int \frac{B}{x-b} \, dx + \int \frac{C}{x-c} \, dx
   \]
   Each of these integrals will yield a natural logarithm function:
   \[
   A \ln|x-a| + B \ln|x-b| + C \ln|x-c| + \text{constant}
   \]

Let me proceed to calculate the values of \( A \), \( B \), and \( C \).It seems that there was a timeout while calculating the constants \( A \), \( B \), and \( C \). I will now manually guide you through the calculation.

### 1. Finding Constants:
By substituting values of \( x = a \), \( x = b \), and \( x = c \), we can solve for \( A \), \( B \), and \( C \).

- When \( x = a \):
  \[
  1 = A(a-b)(a-c)
  \Rightarrow A = \frac{1}{(a-b)(a-c)}
  \]

- When \( x = b \):
  \[
  1 = B(b-a)(b-c)
  \Rightarrow B = \frac{1}{(b-a)(b-c)}
  \]

- When \( x = c \):
  \[
  1 = C(c-a)(c-b)
  \Rightarrow C = \frac{1}{(c-a)(c-b)}
  \]

### 2. Resulting Integral:
The integral can now be expressed as:
\[
\int \frac{1}{(x-a)(x-b)(x-c)} \, dx = \frac{1}{(a-b)(a-c)} \ln|x-a| + \frac{1}{(b-a)(b-c)} \ln|x-b| + \frac{1}{(c-a)(c-b)} \ln|x-c| + C
\]

This is the solution to the given integral.

---

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. How does the method of partial fractions work for different types of polynomials?
2. What are some common mistakes to avoid when solving integrals using partial fractions?
3. How can the technique be extended to higher-order polynomials?
4. What happens if two or more roots in the denominator are repeated?
5. How does this integral apply in real-world scenarios, like electrical engineering?

**Tip:** Always double-check your partial fractions by multiplying them back to verify the original expression!

This article provides a step-by-step solution to the integral ∫ 1 / [(x - a)(x - b)(x - c)] dx using partial fraction decomposition. Learn how to apply logarithmic integration techniques to solve this problem, suitable for advanced high school or college-level students.

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