LARCALC11 4.1.020.
Find the indefinite integral and check your result by differentiation. (Use $C$ for the constant of integration.)
\[
\int\left(\sqrt{x}+\frac{1}{6 \sqrt{x}}\right) d x
\]
Answer
Final Answer: The indefinite integral of the function \(\sqrt{x} + \frac{1}{6\sqrt{x}}\) is \(\boxed{\frac{2}{3}x^{3/2} + \frac{1}{3}x^{1/2} + C}\)
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 \(\boxed{\frac{2}{3}x^{3/2} + \frac{1}{3}x^{1/2} + C}\)
