Step 1 :Given the function \(f(x) = \sqrt{x} + \frac{1}{6\sqrt{x}}\), we are asked to find its indefinite integral.
Step 2 :We calculate the integral of the function to get \(\frac{2}{3}x^{3/2} + \frac{1}{3}x^{1/2} + C\).
Step 3 :To check our result, we differentiate the result of the integral. The derivative is \(\sqrt{x} + \frac{1}{6\sqrt{x}}\), which is the original function. This confirms that our integral is correct.
Step 4 :Final Answer: The indefinite integral of the function \(\sqrt{x} + \frac{1}{6\sqrt{x}}\) is \(\boxed{\frac{2}{3}x^{3/2} + \frac{1}{3}x^{1/2} + C}\)