# Evaluate the integral. int sqrt x(x^3+x/2)dx

Evaluate the integral.
$\int \sqrt{x}\left({x}^{3}+\frac{x}{2}\right)dx$
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Given That
$\int \sqrt{x}\left({x}^{3}+\frac{x}{2}\right)dx$
$=\int \left({x}^{3+\frac{1}{2}}+\frac{{x}^{\frac{1}{2}}\cdot x}{2}\right)dx$
$=\int \left({x}^{\frac{7}{2}}\right)+\frac{{x}^{\frac{3}{2}}}{2}\right)dx$
$=\frac{{x}^{\frac{7}{2}}+1}{\frac{7}{2}+1}+\frac{{x}^{\frac{3}{2}}+1}{\frac{3}{2}+1}+c$
$\frac{2}{9}{x}^{\frac{9}{2}}+\frac{1}{2}\cdot \frac{2}{5}{x}^{\frac{5}{2}}+c$
$=\frac{2}{9}{x}^{\frac{9}{2}}+\frac{1}{5}{x}^{\frac{5}{2}}+c$
$\therefore \int \sqrt{x}\left({x}^{3}+\frac{x}{2}\right)dx=\frac{2}{9}{x}^{\frac{9}{2}}+\frac{1}{5}{x}^{\frac{5}{2}}+c$