determine the following indefinite integral.\n int left(\frac{10}{sqrt{x}}+10sqrt{x}\right)dx \n int…

Explanation:

Step1: Split the integral

$\int\left(\frac{10}{\sqrt{x}}+10\sqrt{x}\right)dx=\int\frac{10}{\sqrt{x}}dx+\int10\sqrt{x}dx$

Step2: Rewrite the terms

$\int\frac{10}{\sqrt{x}}dx = 10\int x^{-\frac{1}{2}}dx$ and $\int10\sqrt{x}dx=10\int x^{\frac{1}{2}}dx$

Step3: Apply power - rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$

$10\int x^{-\frac{1}{2}}dx=10\times\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=10\times\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 20\sqrt{x}$ $10\int x^{\frac{1}{2}}dx=10\times\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}=10\times\frac{x^{\frac{3}{2}}}{\frac{3}{2}}=\frac{20}{3}x^{\frac{3}{2}}$

Step4: Combine the results

$\int\left(\frac{10}{\sqrt{x}}+10\sqrt{x}\right)dx=20\sqrt{x}+\frac{20}{3}x^{\frac{3}{2}}+C$

Answer:

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