evaluate $int_{8}^{27} left(5x^{2}+\frac{sqrt3{x^{2}}}{2}\right)dx.$

Answer:

$\frac{19579}{5}$

Explanation:

Step1: Split the integral

$\int_{8}^{27}(5x^{2}+\frac{\sqrt[3]{x^{2}}}{2})dx=\int_{8}^{27}5x^{2}dx+\int_{8}^{27}\frac{x^{\frac{2}{3}}}{2}dx$

Step2: Integrate term - by - term

For $\int_{8}^{27}5x^{2}dx$, using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $5\times\frac{x^{3}}{3}\big|{8}^{27}=\frac{5}{3}(27^{3}-8^{3})$. For $\int{8}^{27}\frac{x^{\frac{2}{3}}}{2}dx$, using the power rule, we get $\frac{1}{2}\times\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1}\big|_{8}^{27}=\frac{3}{10}\times\frac{1}{2}(27^{\frac{5}{3}}-8^{\frac{5}{3}})$.

Step3: Calculate the definite integrals

$\frac{5}{3}(27^{3}-8^{3})=\frac{5}{3}(19683 - 512)=\frac{5}{3}\times19171=\frac{95855}{3}$. $\frac{3}{20}(27^{\frac{5}{3}}-8^{\frac{5}{3}})=\frac{3}{20}(243 - 32)=\frac{3}{20}\times211=\frac{633}{20}$. $\frac{95855}{3}+\frac{633}{20}=\frac{95855\times20 + 633\times3}{60}=\frac{1917100+1899}{60}=\frac{1918999}{60}=\frac{19579}{5}$.