** Final Answer

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** Step-by-step Solution **

** Specify the solving method

We could not solve this problem by using the method: **Integration by Trigonometric Substitution**

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Rewrite the integrand $\sqrt{x}\left(x+\frac{1}{x}\right)$ in expanded form

Learn how to solve integrals with radicals problems step by step online.

$\int\left(\sqrt{x^{3}}+x^{-\frac{1}{2}}\right)dx$

Learn how to solve integrals with radicals problems step by step online. Integrate int(x^1/2(x+1/x))dx. Rewrite the integrand \sqrt{x}\left(x+\frac{1}{x}\right) in expanded form. Expand the integral \int\left(\sqrt{x^{3}}+x^{-\frac{1}{2}}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\sqrt{x^{3}}dx results in: \frac{2}{5}\sqrt{x^{5}}. The integral \int x^{-\frac{1}{2}}dx results in: 2\sqrt{x}.

** Final Answer

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