The antiderivative of the given expression can be found using the substitution method. We'll make the substitution u = 4 - 9x^2, so du/dx = -18x.
Substituting these into the original expression, we get:
∫{√(4-9x²)/x}dx = ∫{√u/x}du = 2√u * ln|x| + C, where C is an arbitrary constant of integration.
Substituting back for u, we get:
∫{√(4-9x²)/x}dx = 2√(4 - 9x^2) * ln|x| + C.
To verify this result, we can differentiate the antiderivative and see if it matches the original expression. Taking the derivative of the antiderivative with respect to x, we get:
d/dx [2√(4 - 9x^2) * ln|x| + C] = -18x√(4 - 9x^2)/x + 2 * (-9x) / (2√(4 - 9x^2)) = (√(4-9x²)/x)
Thus, the antiderivative found is indeed correct.