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The equation given in your image is an integral that needs to be solved for the value of \( K \). The equation is:

\[
\int_0^1 \left( \frac{2}{10x + 4} - \frac{K}{3x + 4} \right) \, dx = \frac{1}{5} \ln\left(\frac{7}{2}\right) + \frac{5}{3} \ln\left(\frac{4}{7}\right)
\]

### Step-by-step approach:

1. **Evaluate each integral term separately:**

   \[
   \int \frac{2}{10x + 4} \, dx \quad \text{and} \quad \int \frac{K}{3x + 4} \, dx
   \]

   These are integrals of the form:

   \[
   \int \frac{1}{ax + b} \, dx = \frac{1}{a} \ln |ax + b|
   \]

2. **For the first integral:**

   \[
   \int \frac{2}{10x + 4} \, dx = \frac{2}{10} \ln(10x + 4) = \frac{1}{5} \ln(10x + 4)
   \]

3. **For the second integral:**

   \[
   \int \frac{K}{3x + 4} \, dx = \frac{K}{3} \ln(3x + 4)
   \]

4. **Set up the definite integral:**

   Now apply the limits from 0 to 1 for each term.

   For the first term:
   \[
   \left[ \frac{1}{5} \ln(10x + 4) \right]_0^1 = \frac{1}{5} \ln(14) - \frac{1}{5} \ln(4) = \frac{1}{5} \ln\left(\frac{14}{4}\right) = \frac{1}{5} \ln\left(\frac{7}{2}\right)
   \]

   For the second term:
   \[
   \left[ \frac{K}{3} \ln(3x + 4) \right]_0^1 = \frac{K}{3} \ln(7) - \frac{K}{3} \ln(4) = \frac{K}{3} \ln\left(\frac{7}{4}\right)
   \]

5. **Now substitute everything into the equation:**

   \[
   \frac{1}{5} \ln\left(\frac{7}{2}\right) - \frac{K}{3} \ln\left(\frac{7}{4}\right) = \frac{1}{5} \ln\left(\frac{7}{2}\right) + \frac{5}{3} \ln\left(\frac{4}{7}\right)
   \]

6. **Simplify:**

   Notice that the term \(\frac{1}{5} \ln\left(\frac{7}{2}\right)\) cancels out on both sides of the equation, so we are left with:

   \[
   - \frac{K}{3} \ln\left(\frac{7}{4}\right) = \frac{5}{3} \ln\left(\frac{4}{7}\right)
   \]

7. **Solving for \( K \):**

   We know that:

   \[
   \ln\left(\frac{4}{7}\right) = - \ln\left(\frac{7}{4}\right)
   \]

   So the equation becomes:



Consider the equation: \[ \int_0^1 \left( \frac{2}{10x + 4} - \frac{K}{3x + 4} \right) \, dx = \frac{1}{5} \ln\left(\frac{7}{2}\right) + \frac{5}{3} \ln\left(\frac{4}{7}\right) \]

Learn how to solve the integral equation \( \int_0^1 \left( \frac{2}{10x + 4} - \frac{K}{3x + 4} \right) dx \) to find the value of K. This step-by-step solution covers the key concepts of logarithmic integration and definite integrals, suitable for Calculus II students.

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