Calculate the indefinite integral \[ \int \frac{4 d x}{\sqrt{36-9 x^{2}}} \]

Step 1 ：We are given the integral \(\int \frac{4 dx}{\sqrt{36-9 x^{2}}}\).

Step 2 ：This integral is in the form of \(\int \frac{a dx}{\sqrt{b^2 - a^2 x^2}}\), which is equal to \(\frac{1}{a} \sin^{-1}(\frac{a x}{b}) + C\).

Step 3 ：In our case, a = 3, b = 6 and the constant multiplier is 4.

Step 4 ：We substitute these values into the formula to get the result.

Step 5 ：The result is given in piecewise form, which means the result is different depending on the value of x. However, since we're dealing with real numbers, we can ignore the part with the imaginary unit 'I'.

Step 6 ：The final result is \(\frac{4}{3} \sin^{-1}(\frac{x}{2}) + C\).

Step 7 ：\(\boxed{\frac{4}{3} \sin^{-1}\left(\frac{x}{2}\right) + C}\) is the final answer.