7 (a) Express \( \frac{4x+3}{(x-1)^2} \) in the form \( \frac{A}{x-1} + \frac{B}{(x-1)^2} \) - AQA - A-Level Maths Pure - Question 7 - 2019 - Paper 3

Integrate ( \frac{4x+3}{(x-1)^2} ):

Using the earlier expression, we can integrate:

$\int \frac{4}{x-1} + \frac{7}{(x-1)^2} dx$

This gives:

$\int \frac{4}{x-1} dx + \int \frac{7}{(x-1)^2} dx = 4\ln |x-1| - \frac{7}{x-1} + C$

Evaluating from 3 to 4:

$\left[ 4\ln |4-1| - \frac{7}{4-1} \right] - \left[ 4\ln |3-1| - \frac{7}{3-1} \right]$

Substituting values, we get:

$= 4\ln 3 - \frac{7}{3} - (4\ln 2 - \frac{7}{2})$

Combining terms gives us:

$= 4\ln(\frac{3}{2}) + \frac{7}{2} - \frac{7}{3}$.

By simplifying further, this shows the required form ( p + \ln q ).