Let the sum of all solutions of the equation `3sqrt(3) sin^(3)x + cos^(3)x + 3sqrt(3) sinx cosx=1` in the interval `[0,10pi]` denoted by S. Then `(3S)/pi` is………………..

Explanation

To solve the equation \( 3\sqrt{3} \sin^3 x + \cos^3 x + 3\sqrt{3} \sin x \cos x = 1 \) and find the sum of all solutions in the interval \([0, 10\pi]\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3\sqrt{3} \sin^3 x + \cos^3 x + 3\sqrt{3} \sin x \cos x = 1 \]
We can express \(3\sqrt{3} \sin^3 x\) as \((\sqrt{3} \sin x)^3\) and \(3\sqrt{3} \sin x \cos x\) as \(3(\sqrt{3} \sin x)(\cos x)\). This allows us to rewrite the equation in the form of a sum of cubes:
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