Question 4
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Use the table of integration formulas to identify and use an appropriate formula to find the following indefinite integral:
\[
\int x(4 x+9)^{-2} d x
\]
$\frac{1}{16}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c$
$\frac{1}{16}\left(\ln |4 x+9|+\frac{4}{4 x+9}\right)+c$
$\frac{1}{4}\left(\ln |4 x+9|+\frac{4}{4 x+9}\right)+c$
$\frac{1}{4}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c$

Answer

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Answer

The indefinite integral of $\int x(4 x+9)^{-2} d x$ is $\boxed{\frac{1}{16}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c}$

Steps

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 $\boxed{\frac{1}{16}\left(\ln |4 x+9|+\frac{9}{4 x+9}\right)+c}$