https://file.aiquickdraw.com/mathbot/file/1727539779550vq0phlwt.jpg

The image contains an integral expression that appears to be:

\[
\int \frac{dx}{(x+2)(x+3)}
\]

This is a standard integral that can be solved using partial fraction decomposition.

### Steps for solving:

1. **Rewrite the integrand using partial fractions**:
   We aim to express the integrand as a sum of simpler fractions:
   \[
   \frac{1}{(x+2)(x+3)} = \frac{A}{x+2} + \frac{B}{x+3}
   \]

2. **Solve for \( A \) and \( B \)**:
   Multiply both sides by \((x+2)(x+3)\):
   \[
   1 = A(x+3) + B(x+2)
   \]
   Expanding this:
   \[
   1 = A(x) + 3A + B(x) + 2B
   \]
   Grouping the terms:
   \[
   1 = (A + B)x + (3A + 2B)
   \]
   Now, equate the coefficients of \(x\) and the constant term:
   \[
   A + B = 0
   \]
   \[
   3A + 2B = 1
   \]

3. **Solve the system of equations**:
   From \(A + B = 0\), we get \(B = -A\). Substitute this into the second equation:
   \[
   3A + 2(-A) = 1 \quad \Rightarrow \quad 3A - 2A = 1 \quad \Rightarrow \quad A = 1
   \]
   Now, \(B = -A = -1\).

4. **Rewrite the integral**:
   Using the values of \(A\) and \(B\), the integral becomes:
   \[
   \int \frac{1}{(x+2)(x+3)} \, dx = \int \left( \frac{1}{x+2} - \frac{1}{x+3} \right) dx
   \]

5. **Integrate each term**:
   \[
   \int \left( \frac{1}{x+2} - \frac{1}{x+3} \right) dx = \ln|x+2| - \ln|x+3| + C
   \]

6. **Simplify**:
   \[
   \ln\left(\frac{|x+2|}{|x+3|}\right) + C
   \]

Thus, the solution to the integral is:

\[
\ln\left(\frac{|x+2|}{|x+3|}\right) + C
\]

Would you like more details on any of the steps, or do you have any questions?

### Related Questions:
1. How is partial fraction decomposition applied to more complex rational functions?
2. What are the conditions under which partial fraction decomposition can be used?
3. How would you integrate if the denominator had a quadratic factor?
4. How does the natural logarithm function behave for negative values?
5. What happens when the limits of integration are added to this problem?

### Tip:
Always check if you can simplify the integrand using techniques like partial fractions or trigonometric identities before proceeding with integration.

Learn how to solve the integral ∫ dx / ((x+2)(x+3)) using partial fraction decomposition. The solution involves breaking the fraction into simpler terms and applying logarithmic integration. Suitable for students in Grades 11-12 or early undergraduate studies.

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