Find the value of \[\int_{2}^{2} \frac{6x + 1}{6x^{2} - 7x + 2} \; dx,\] expressing your answer in the form \(mln 2 + nln 3\), where \(m\) and \(n\) are integers. - AQA - A-Level Maths Mechanics - Question 6 - 2019 - Paper 3

Expanding:

[6x + 1 = A(2x - 2) + B(3x - 1)]

Setting (x = 1) will help us find (A) and (B):

Solving gives:

• When (x = 1): (6(1) + 1 = A(0) + B(3(1) - 1)) leads to finding (B).

• Additionally, set up another equation using another value for (x) to find (A).

For the values of (A) and (B), use substitution to get:

• Final partial fraction form is obtained from constants derived.

Now, integrate the resulting fractions:

[\int \left( \frac{A}{3x - 1} + \frac{B}{2x - 2} \right)dx]

Apply integral rules, yielding:

[A \ln |3x - 1| + B \ln |2x - 2| + C]

Apply the limits from 2 to 2:

[A \ln |3(2) - 1| + B \ln |2(2) - 2| - (A \ln |3(2) - 1| + B \ln |2(2) - 2|)]

The result in the form (mln 2 + nln 3) will yield integer values for (m) and (n) when simplifying the logarithmic results.