# Find the following integral. \int \sqrt{x}(x^{3}+\frac{x}{2})dx

Find the following integral.
$\int \sqrt{x}\left({x}^{3}+\frac{x}{2}\right)dx$
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Raheem Donnelly
Given that
$\int \sqrt{x}\left({x}^{3}+\frac{x}{2}\right)dx$
$=\int \left({x}^{3+\frac{1}{2}}+\frac{{x}^{\frac{1}{2}\cdot x}}{2}\right)dx$
$=\int \left({x}^{\frac{7}{2}}+\frac{{x}^{\frac{3}{2}}}{2}\right)dx$
$=\frac{{x}^{\frac{7}{2}+1}}{\frac{7}{2}+1}+\frac{{x}^{\frac{3}{2}+1}}{\frac{\frac{3}{2}+1}{2}}+C$
$=\frac{2}{9}{x}^{\frac{9}{2}}+\frac{1}{2}\cdot \frac{2}{5}\cdot {x}^{\frac{5}{2}}+C$
$=\frac{2}{9}{x}^{\frac{9}{2}}+\frac{1}{5}\cdot {x}^{\frac{5}{2}}+C$
$\therefore \int \sqrt{x}\left({x}^{3}+\frac{x}{2}\right)dx=\frac{2}{9}\cdot {x}^{\frac{9}{2}}+\frac{1}{5}\cdot {x}^{\frac{5}{2}}+C$