match the integral with the rewritten integral\n∫(2√x⁵/3) dx\n∫(2x¹/²) dx\n∫(2√x³) dx\n∫(2√x) dx\n∫(√x)…

Explanation:

Step1: Recall the rule $\sqrt{x^n}=x^{\frac{n}{2}}$

For $\int\left(\frac{2\sqrt{x^5}}{3}\right)dx$, since $\sqrt{x^5}=x^{\frac{5}{2}}$, it can be rewritten as $\int\left(\frac{2x^{\frac{5}{2}}}{3}\right)dx$.

Step2: For $\int(2\sqrt{x})dx$

Since $\sqrt{x}=x^{\frac{1}{2}}$, it can be rewritten as $\int(2x^{\frac{1}{2}})dx$.

Step3: For $\int(2\sqrt{x^3})dx$

Since $\sqrt{x^3}=x^{\frac{3}{2}}$, it can be rewritten as $\int(2x^{\frac{3}{2}})dx$.

Step4: For $\int(\sqrt{x})dx$

Since $\sqrt{x}=x^{\frac{1}{2}}$, it can be rewritten as $\int(x^{\frac{1}{2}})dx$.

Answer:

$\int\left(\frac{2\sqrt{x^5}}{3}\right)dx$ matches $\int\left(\frac{2x^{\frac{5}{2}}}{3}\right)dx$; $\int(2\sqrt{x})dx$ matches $\int(2x^{\frac{1}{2}})dx$; $\int(2\sqrt{x^3})dx$ matches $\int(2x^{\frac{3}{2}})dx$; $\int(\sqrt{x})dx$ matches $\int(x^{\frac{1}{2}})dx$