find the most general antiderivative or indefinite integral. ∫(10/√x + 10√x) dx ∫(10/√x + 10√x) dx = □

Explanation:

Step1: Rewrite the integrand

Rewrite $\frac{10}{\sqrt{x}}$ as $10x^{-\frac{1}{2}}$ and $10\sqrt{x}$ as $10x^{\frac{1}{2}}$. So the integral becomes $\int(10x^{-\frac{1}{2}} + 10x^{\frac{1}{2}})dx$.

Step2: Use the power - rule for integration

The power - rule for integration is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int10x^{-\frac{1}{2}}dx$, we have $10\times\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=10\times\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 20x^{\frac{1}{2}}$. For $\int10x^{\frac{1}{2}}dx$, we have $10\times\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=10\times\frac{x^{\frac{3}{2}}}{\frac{3}{2}}=\frac{20}{3}x^{\frac{3}{2}}$.

Step3: Combine the results

$\int(10x^{-\frac{1}{2}} + 10x^{\frac{1}{2}})dx=20x^{\frac{1}{2}}+\frac{20}{3}x^{\frac{3}{2}}+C$.

Answer:

$20\sqrt{x}+\frac{20}{3}x^{\frac{3}{2}}+C$