Problem
Question 4 Mark this question Use the table of integration formulas to identify and use an appropriate formula to find the following indefinite integral: \[ \int x(4 x+9)^{-2} d x \] $\frac{1}{16}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c$ $\frac{1}{16}\left(\ln |4 x+9|+\frac{4}{4 x+9}\right)+c$ $\frac{1}{4}\left(\ln |4 x+9|+\frac{4}{4 x+9}\right)+c$ $\frac{1}{4}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c$
Solution
Step 1 ：Given the integral \(\int x(4 x+9)^{-2} d x\)
Step 2 ：Let \(u = 4x + 9\)
Step 3 ：Then, \(du = 4dx\). So, \(dx = du/4\)
Step 4 ：Substitute \(u\) and \(dx\) into the integral, we get \(\int \frac{1}{4}u^{-2} du\)
Step 5 ：This is a standard integral and can be solved easily
Step 6 ：The indefinite integral of \(\int x(4 x+9)^{-2} d x\) is \(\boxed{\frac{1}{16}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c}\)
