Integrals Of Some Particular Functions

Step 1: Partial Fraction Decomposition

We need to decompose the integrand into partial fractions:

Multiply both sides by :

Step 2: Solve for A, B, and C

* For :

* For :

* For :

Step 3: Rewrite the Integral

Now we can rewrite the integral as:

Step 4: Evaluate the Integrals

Step 5: Combine Logarithms

Now, let's rewrite to match the given form:

By direct comparison

Conclusion:

The value of k is 12.