Problem

$\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x$

Answer

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Answer

Final Answer: The integral of the function \(\sqrt{x}-\frac{1}{\sqrt{x}}\) with respect to \(x\) is \(\boxed{\frac{2}{3}x^{3/2} - 2\sqrt{x}}\)

Steps

Step 1 ：Given the integral function \(\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) dx\)

Step 2 ：We can split this into two separate integrals: \(\int\sqrt{x} dx\) and \(-\int\frac{1}{\sqrt{x}} dx\)

Step 3 ：The integral of \(\sqrt{x}\) is \(\frac{2}{3}x^{3/2}\)

Step 4 ：The integral of \(\frac{1}{\sqrt{x}}\) is \(2\sqrt{x}\)

Step 5 ：So, the integral of the given function is \(\frac{2}{3}x^{3/2} - 2\sqrt{x}\)

Step 6 ：Final Answer: The integral of the function \(\sqrt{x}-\frac{1}{\sqrt{x}}\) with respect to \(x\) is \(\boxed{\frac{2}{3}x^{3/2} - 2\sqrt{x}}\)

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