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The given problem asks to find the real part of the following expression:

\[
\frac{3}{k+i} - \frac{i}{k-i}
\]

where \( k \) is a real number and \( i \) is the imaginary unit.

### Step 1: Simplify each term individually

**First term**:
\[
\frac{3}{k+i}
\]
To simplify this, multiply both the numerator and denominator by the complex conjugate of the denominator, \( k-i \):
\[
\frac{3}{k+i} \cdot \frac{k-i}{k-i} = \frac{3(k-i)}{(k+i)(k-i)} = \frac{3(k-i)}{k^2 + 1}
\]
Thus, 
\[
\frac{3}{k+i} = \frac{3k}{k^2+1} - \frac{3i}{k^2+1}
\]

**Second term**:
\[
\frac{i}{k-i}
\]
Multiply both the numerator and denominator by the complex conjugate of the denominator, \( k+i \):
\[
\frac{i}{k-i} \cdot \frac{k+i}{k+i} = \frac{i(k+i)}{(k-i)(k+i)} = \frac{i(k+i)}{k^2 + 1} = \frac{ik - 1}{k^2+1}
\]
Thus,
\[
\frac{i}{k-i} = \frac{-1}{k^2+1} + \frac{ik}{k^2+1}
\]

### Step 2: Subtract the two expressions

Now, subtract the second term from the first:
\[
\frac{3k}{k^2+1} - \frac{3i}{k^2+1} - \left( \frac{-1}{k^2+1} + \frac{ik}{k^2+1} \right)
\]
Distribute the minus sign:
\[
\frac{3k}{k^2+1} - \frac{3i}{k^2+1} + \frac{1}{k^2+1} - \frac{ik}{k^2+1}
\]
Combine real and imaginary parts:
- Real part: \( \frac{3k}{k^2+1} + \frac{1}{k^2+1} = \frac{3k+1}{k^2+1} \)
- Imaginary part: \( -\frac{3i}{k^2+1} - \frac{ik}{k^2+1} = -\frac{i(3+k)}{k^2+1} \)

### Step 3: Identify the real part

The real part of the expression is:
\[
\frac{3k+1}{k^2+1}
\]

Thus, the real part of the given expression is \( \boxed{\frac{3k+1}{k^2+1}} \).

Let me know if you want further details or have any questions.

Here are some related questions:
1. What is the complex conjugate of a complex number?
2. How do you divide two complex numbers?
3. What does the real part of a complex number represent?
4. Can you solve for the imaginary part of the original expression?
5. How would this change if \( k \) were a complex number?

**Tip**: When dealing with complex fractions, always multiply by the conjugate to simplify and separate real and imaginary parts!

If k is a real number, then the real part of 3/(k+i) - i/(k-i) is?

Learn how to find the real part of the complex expression 3/(k+i) - i/(k-i), where k is a real number. This step-by-step solution involves simplifying complex fractions and identifying the real and imaginary components. Suitable for students in Grades 10-12.

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